| L(s) = 1 | − 2·4-s + 5-s + 2·7-s + 3·11-s − 4·13-s + 4·16-s + 6·17-s − 19-s − 2·20-s + 6·23-s + 25-s − 4·28-s + 9·29-s − 31-s + 2·35-s + 8·37-s − 3·41-s − 4·43-s − 6·44-s − 12·47-s − 3·49-s + 8·52-s − 6·53-s + 3·55-s − 3·59-s − 10·61-s − 8·64-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s + 0.755·7-s + 0.904·11-s − 1.10·13-s + 16-s + 1.45·17-s − 0.229·19-s − 0.447·20-s + 1.25·23-s + 1/5·25-s − 0.755·28-s + 1.67·29-s − 0.179·31-s + 0.338·35-s + 1.31·37-s − 0.468·41-s − 0.609·43-s − 0.904·44-s − 1.75·47-s − 3/7·49-s + 1.10·52-s − 0.824·53-s + 0.404·55-s − 0.390·59-s − 1.28·61-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.338654780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.338654780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−11.29810456021436859378746330171, −10.04624014100845215666127682432, −9.562964346295115144850959285425, −8.538687880420415041094117740367, −7.73102144096255023104365276788, −6.47289908140934830552102870516, −5.18690236062207152876422818521, −4.59634975877424853850281958730, −3.14233849758258863442381605357, −1.27645214139714047950709921081,
1.27645214139714047950709921081, 3.14233849758258863442381605357, 4.59634975877424853850281958730, 5.18690236062207152876422818521, 6.47289908140934830552102870516, 7.73102144096255023104365276788, 8.538687880420415041094117740367, 9.562964346295115144850959285425, 10.04624014100845215666127682432, 11.29810456021436859378746330171
