| L(s) = 1 | + 2.79·3-s − 3·5-s + 7-s + 4.79·9-s + 3.79·11-s + 13-s − 8.37·15-s + 3.79·17-s − 19-s + 2.79·21-s + 4.58·23-s + 4·25-s + 4.99·27-s − 3.79·29-s + 7.37·31-s + 10.5·33-s − 3·35-s − 5·37-s + 2.79·39-s − 3.79·41-s − 2·43-s − 14.3·45-s + 10.5·47-s + 49-s + 10.5·51-s + 8.37·53-s − 11.3·55-s + ⋯ |
| L(s) = 1 | + 1.61·3-s − 1.34·5-s + 0.377·7-s + 1.59·9-s + 1.14·11-s + 0.277·13-s − 2.16·15-s + 0.919·17-s − 0.229·19-s + 0.609·21-s + 0.955·23-s + 0.800·25-s + 0.962·27-s − 0.704·29-s + 1.32·31-s + 1.84·33-s − 0.507·35-s − 0.821·37-s + 0.446·39-s − 0.592·41-s − 0.304·43-s − 2.14·45-s + 1.54·47-s + 0.142·49-s + 1.48·51-s + 1.15·53-s − 1.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.694053126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.694053126\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 11 | \( 1 - 3.79T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 23 | \( 1 - 4.58T + 23T^{2} \) |
| 29 | \( 1 + 3.79T + 29T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + 3.79T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 8.37T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 9.37T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 7.58T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.899059610424442737721966313960, −7.27806321378636324971419931553, −6.82046336853401144973704014953, −5.69062052750856939954129073537, −4.60546983234004483835077327838, −4.03128320830289887892689200710, −3.48273773988637319191263379962, −2.90904162589133298160690701316, −1.81028748186173786243750636227, −0.918274648196623251441518733459,
0.918274648196623251441518733459, 1.81028748186173786243750636227, 2.90904162589133298160690701316, 3.48273773988637319191263379962, 4.03128320830289887892689200710, 4.60546983234004483835077327838, 5.69062052750856939954129073537, 6.82046336853401144973704014953, 7.27806321378636324971419931553, 7.899059610424442737721966313960
