| L(s) = 1 | + 5-s + 3.74·7-s + 2.54·11-s + 13-s − 1.74·17-s − 2.29·19-s − 1.74·23-s + 25-s + 4.29·29-s + 2·31-s + 3.74·35-s + 8.03·37-s + 2.94·41-s + 7.49·43-s − 3.49·47-s + 7.03·49-s − 2.54·53-s + 2.54·55-s − 8.58·59-s − 4.03·61-s + 65-s − 1.49·67-s + 13.5·71-s + 9.78·73-s + 9.52·77-s + 8.94·79-s − 1.74·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.41·7-s + 0.766·11-s + 0.277·13-s − 0.423·17-s − 0.525·19-s − 0.364·23-s + 0.200·25-s + 0.796·29-s + 0.359·31-s + 0.633·35-s + 1.32·37-s + 0.460·41-s + 1.14·43-s − 0.509·47-s + 1.00·49-s − 0.349·53-s + 0.342·55-s − 1.11·59-s − 0.516·61-s + 0.124·65-s − 0.182·67-s + 1.60·71-s + 1.14·73-s + 1.08·77-s + 1.00·79-s − 0.189·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.821207962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.821207962\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 - 2.54T + 11T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 19 | \( 1 + 2.29T + 19T^{2} \) |
| 23 | \( 1 + 1.74T + 23T^{2} \) |
| 29 | \( 1 - 4.29T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 8.03T + 37T^{2} \) |
| 41 | \( 1 - 2.94T + 41T^{2} \) |
| 43 | \( 1 - 7.49T + 43T^{2} \) |
| 47 | \( 1 + 3.49T + 47T^{2} \) |
| 53 | \( 1 + 2.54T + 53T^{2} \) |
| 59 | \( 1 + 8.58T + 59T^{2} \) |
| 61 | \( 1 + 4.03T + 61T^{2} \) |
| 67 | \( 1 + 1.49T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 9.78T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 6.94T + 89T^{2} \) |
| 97 | \( 1 - 5.34T + 97T^{2} \) |
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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
−8.123828467093779490125728468502, −7.86725954399733558223681678605, −6.69501728307721033115612288607, −6.22040114031828792475551091570, −5.31574569545348030193321253032, −4.53810384897558958391001309684, −4.00757299596852029941284387699, −2.70624800934768997680029524129, −1.85856756323702948503405399687, −1.00609785835314980285192735498,
1.00609785835314980285192735498, 1.85856756323702948503405399687, 2.70624800934768997680029524129, 4.00757299596852029941284387699, 4.53810384897558958391001309684, 5.31574569545348030193321253032, 6.22040114031828792475551091570, 6.69501728307721033115612288607, 7.86725954399733558223681678605, 8.123828467093779490125728468502
