| L(s) = 1 | + 35.0·7-s − 25.6·11-s − 37.6·13-s − 95.7·17-s + 50.8·19-s + 110.·23-s + 54.5·29-s + 198.·31-s − 266.·37-s − 103.·41-s − 108·43-s + 597.·47-s + 887.·49-s − 305.·53-s + 223.·59-s + 485.·61-s + 876.·67-s − 585.·71-s + 1.13e3·73-s − 899.·77-s + 685.·79-s + 305.·83-s − 887.·89-s − 1.32e3·91-s + 556.·97-s − 1.59e3·101-s − 1.35e3·103-s + ⋯ |
| L(s) = 1 | + 1.89·7-s − 0.702·11-s − 0.803·13-s − 1.36·17-s + 0.614·19-s + 1.00·23-s + 0.349·29-s + 1.14·31-s − 1.18·37-s − 0.395·41-s − 0.383·43-s + 1.85·47-s + 2.58·49-s − 0.792·53-s + 0.493·59-s + 1.01·61-s + 1.59·67-s − 0.978·71-s + 1.82·73-s − 1.33·77-s + 0.975·79-s + 0.404·83-s − 1.05·89-s − 1.52·91-s + 0.582·97-s − 1.56·101-s − 1.29·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.570275207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.570275207\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 35.0T + 343T^{2} \) |
| 11 | \( 1 + 25.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 95.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 50.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 110.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 54.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 198.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 266.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 103.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 108T + 7.95e4T^{2} \) |
| 47 | \( 1 - 597.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 305.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 223.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 485.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 876.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 585.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 685.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 305.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 887.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 556.T + 9.12e5T^{2} \) |
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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
−8.688414636296261576290979469402, −8.209161153596871372877784083059, −7.39748835658298214572321304864, −6.71705555981179115729392254027, −5.27171269892024354760214251897, −5.01244367438998317976151494092, −4.14631113891326389578988216549, −2.69835740062118030375420233112, −1.94006380646689545956139097520, −0.76345755174280897898008155279,
0.76345755174280897898008155279, 1.94006380646689545956139097520, 2.69835740062118030375420233112, 4.14631113891326389578988216549, 5.01244367438998317976151494092, 5.27171269892024354760214251897, 6.71705555981179115729392254027, 7.39748835658298214572321304864, 8.209161153596871372877784083059, 8.688414636296261576290979469402
