| L(s) = 1 | − 228·3-s + 666·5-s + 1.96e4·9-s + 3.04e4·11-s − 6.46e4·13-s − 1.51e5·15-s − 5.90e5·17-s − 3.46e4·19-s − 1.04e6·23-s + 1.95e6·25-s − 1.61e6·27-s + 8.81e6·29-s + 7.40e6·31-s − 6.93e6·33-s − 1.02e7·37-s + 1.47e7·39-s + 3.67e7·41-s − 5.04e5·43-s + 1.31e7·45-s + 4.95e7·47-s + 1.34e8·51-s + 6.63e7·53-s + 2.02e7·55-s + 7.90e6·57-s + 6.15e7·59-s − 3.56e7·61-s − 4.30e7·65-s + ⋯ |
| L(s) = 1 | − 1.62·3-s + 0.476·5-s + 9-s + 0.626·11-s − 0.628·13-s − 0.774·15-s − 1.71·17-s − 0.0610·19-s − 0.781·23-s + 25-s − 0.583·27-s + 2.31·29-s + 1.43·31-s − 1.01·33-s − 0.897·37-s + 1.02·39-s + 2.02·41-s − 0.0225·43-s + 0.476·45-s + 1.48·47-s + 2.78·51-s + 1.15·53-s + 0.298·55-s + 0.0992·57-s + 0.661·59-s − 0.329·61-s − 0.299·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.975654400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.975654400\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 76 p T + 3589 p^{2} T^{2} + 76 p^{10} T^{3} + p^{18} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 666 T - 1509569 T^{2} - 666 p^{9} T^{3} + p^{18} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 30420 T - 1432571291 T^{2} - 30420 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 32338 T + p^{9} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 590994 T + 230686031539 T^{2} + 590994 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 34676 T - 321485272803 T^{2} + 34676 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 1048536 T - 701724918167 T^{2} + 1048536 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4409406 T + p^{9} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 7401184 T + 28337902441185 T^{2} - 7401184 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 10234502 T - 25216708607073 T^{2} + 10234502 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 18352746 T + p^{9} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 252340 T + p^{9} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 49517136 T + 1332816284539729 T^{2} - 49517136 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 66396906 T + 1108785534570703 T^{2} - 66396906 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 61523748 T - 4877824250687435 T^{2} - 61523748 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 35638622 T - 10424034714775257 T^{2} + 35638622 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 181742372 T + 5823755383891437 T^{2} + 181742372 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 90904968 T + p^{9} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 262978678 T + 10286198374359771 T^{2} - 262978678 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 116502832 T - 106278686118598095 T^{2} - 116502832 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9563724 T + p^{9} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 611826714 T + 23975524256552587 T^{2} + 611826714 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 259312798 T + p^{9} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.92885158683998848690320456234, −10.75335532318475153533060713557, −10.32555116267252232845906795820, −9.642128856924937169747340041343, −9.286018628773007293070383461082, −8.530415950199590885272733694587, −8.263985450927726004990980026538, −7.24746930420000248706995066983, −6.77554037916555869637461693035, −6.49531502461069077966208174344, −5.89160763118829976715389922641, −5.56365806197580868428675299401, −4.65298807266619584120924875624, −4.58635887526713798532705816180, −3.87888115194624231126762499464, −2.65945268162743583615632631211, −2.44863574370038809959705020693, −1.51794658397927554468257978903, −0.64927693097544580724205138325, −0.56630292061632817830660247961,
0.56630292061632817830660247961, 0.64927693097544580724205138325, 1.51794658397927554468257978903, 2.44863574370038809959705020693, 2.65945268162743583615632631211, 3.87888115194624231126762499464, 4.58635887526713798532705816180, 4.65298807266619584120924875624, 5.56365806197580868428675299401, 5.89160763118829976715389922641, 6.49531502461069077966208174344, 6.77554037916555869637461693035, 7.24746930420000248706995066983, 8.263985450927726004990980026538, 8.530415950199590885272733694587, 9.286018628773007293070383461082, 9.642128856924937169747340041343, 10.32555116267252232845906795820, 10.75335532318475153533060713557, 10.92885158683998848690320456234
