| L(s) = 1 | + 1.61·3-s + 4.23·5-s + 7-s − 0.381·9-s − 2.61·11-s − 3.47·13-s + 6.85·15-s − 5.85·17-s + 19-s + 1.61·21-s + 8.23·23-s + 12.9·25-s − 5.47·27-s + 4.61·29-s + 3.85·31-s − 4.23·33-s + 4.23·35-s + 8.23·37-s − 5.61·39-s + 11.5·41-s + 4.47·43-s − 1.61·45-s − 7.47·47-s + 49-s − 9.47·51-s − 6.09·53-s − 11.0·55-s + ⋯ |
| L(s) = 1 | + 0.934·3-s + 1.89·5-s + 0.377·7-s − 0.127·9-s − 0.789·11-s − 0.962·13-s + 1.76·15-s − 1.41·17-s + 0.229·19-s + 0.353·21-s + 1.71·23-s + 2.58·25-s − 1.05·27-s + 0.857·29-s + 0.692·31-s − 0.737·33-s + 0.716·35-s + 1.35·37-s − 0.899·39-s + 1.80·41-s + 0.681·43-s − 0.241·45-s − 1.08·47-s + 0.142·49-s − 1.32·51-s − 0.836·53-s − 1.49·55-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
| \(L(1)\) |
\(\approx\) |
\(4.189625279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.189625279\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 - 4.23T + 5T^{2} \) |
| 11 | \( 1 + 2.61T + 11T^{2} \) |
| 13 | \( 1 + 3.47T + 13T^{2} \) |
| 17 | \( 1 + 5.85T + 17T^{2} \) |
| 23 | \( 1 - 8.23T + 23T^{2} \) |
| 29 | \( 1 - 4.61T + 29T^{2} \) |
| 31 | \( 1 - 3.85T + 31T^{2} \) |
| 37 | \( 1 - 8.23T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 + 7.47T + 47T^{2} \) |
| 53 | \( 1 + 6.09T + 53T^{2} \) |
| 59 | \( 1 - 11T + 59T^{2} \) |
| 61 | \( 1 - 4.23T + 61T^{2} \) |
| 67 | \( 1 + 7.56T + 67T^{2} \) |
| 71 | \( 1 - 3.76T + 71T^{2} \) |
| 73 | \( 1 - 3.61T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 3.47T + 97T^{2} \) |
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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
−7.889763638627080904797154722078, −7.07666420384606181151944598699, −6.41597735150515538066262257814, −5.68564352726399266896976539462, −4.98482593387121022959827339150, −4.49207863582566691064157493999, −2.99929835083547694025198292649, −2.52451419717471534028990093556, −2.16272039062793630980081983116, −0.960471341176697098285565153514,
0.960471341176697098285565153514, 2.16272039062793630980081983116, 2.52451419717471534028990093556, 2.99929835083547694025198292649, 4.49207863582566691064157493999, 4.98482593387121022959827339150, 5.68564352726399266896976539462, 6.41597735150515538066262257814, 7.07666420384606181151944598699, 7.889763638627080904797154722078
