| L(s) = 1 | − 2·4-s + 2·5-s + 2·7-s − 11-s − 5·13-s + 4·16-s − 3·17-s − 4·20-s + 6·23-s − 25-s − 4·28-s + 10·29-s − 10·31-s + 4·35-s + 10·41-s − 6·43-s + 2·44-s − 2·47-s − 3·49-s + 10·52-s − 5·53-s − 2·55-s + 5·59-s − 8·64-s − 10·65-s − 10·67-s + 6·68-s + ⋯ |
| L(s) = 1 | − 4-s + 0.894·5-s + 0.755·7-s − 0.301·11-s − 1.38·13-s + 16-s − 0.727·17-s − 0.894·20-s + 1.25·23-s − 1/5·25-s − 0.755·28-s + 1.85·29-s − 1.79·31-s + 0.676·35-s + 1.56·41-s − 0.914·43-s + 0.301·44-s − 0.291·47-s − 3/7·49-s + 1.38·52-s − 0.686·53-s − 0.269·55-s + 0.650·59-s − 64-s − 1.24·65-s − 1.22·67-s + 0.727·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.02141796390654, −14.43731599330359, −14.25460990325601, −13.65978356744097, −13.09029854213355, −12.67736499430158, −12.25044791440722, −11.39185897534930, −10.92543403311972, −10.32012381648904, −9.722031488530785, −9.413317211608500, −8.816745360184672, −8.270224734224267, −7.676856977360689, −7.080843637111426, −6.400777479439464, −5.639037846598559, −5.018414608006462, −4.854812299967249, −4.153283438851151, −3.225568105264048, −2.496433810684977, −1.860349822486333, −0.9852191847912993, 0,
0.9852191847912993, 1.860349822486333, 2.496433810684977, 3.225568105264048, 4.153283438851151, 4.854812299967249, 5.018414608006462, 5.639037846598559, 6.400777479439464, 7.080843637111426, 7.676856977360689, 8.270224734224267, 8.816745360184672, 9.413317211608500, 9.722031488530785, 10.32012381648904, 10.92543403311972, 11.39185897534930, 12.25044791440722, 12.67736499430158, 13.09029854213355, 13.65978356744097, 14.25460990325601, 14.43731599330359, 15.02141796390654
