| L(s) = 1 | + 0.301·2-s − 1.06·3-s − 1.90·4-s − 5-s − 0.320·6-s − 2.50·7-s − 1.17·8-s − 1.87·9-s − 0.301·10-s + 2.44·11-s + 2.02·12-s − 5.61·13-s − 0.757·14-s + 1.06·15-s + 3.46·16-s − 0.565·18-s − 7.13·19-s + 1.90·20-s + 2.66·21-s + 0.737·22-s − 0.860·23-s + 1.25·24-s + 25-s − 1.69·26-s + 5.17·27-s + 4.79·28-s + 3.75·29-s + ⋯ |
| L(s) = 1 | + 0.213·2-s − 0.612·3-s − 0.954·4-s − 0.447·5-s − 0.130·6-s − 0.948·7-s − 0.416·8-s − 0.624·9-s − 0.0954·10-s + 0.737·11-s + 0.584·12-s − 1.55·13-s − 0.202·14-s + 0.273·15-s + 0.865·16-s − 0.133·18-s − 1.63·19-s + 0.426·20-s + 0.581·21-s + 0.157·22-s − 0.179·23-s + 0.255·24-s + 0.200·25-s − 0.332·26-s + 0.995·27-s + 0.905·28-s + 0.696·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4056371560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4056371560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.301T + 2T^{2} \) |
| 3 | \( 1 + 1.06T + 3T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 5.61T + 13T^{2} \) |
| 19 | \( 1 + 7.13T + 19T^{2} \) |
| 23 | \( 1 + 0.860T + 23T^{2} \) |
| 29 | \( 1 - 3.75T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 + 5.10T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 + 2.62T + 43T^{2} \) |
| 47 | \( 1 + 2.30T + 47T^{2} \) |
| 53 | \( 1 - 2.77T + 53T^{2} \) |
| 59 | \( 1 - 7.44T + 59T^{2} \) |
| 61 | \( 1 - 0.906T + 61T^{2} \) |
| 67 | \( 1 - 6.69T + 67T^{2} \) |
| 71 | \( 1 + 0.240T + 71T^{2} \) |
| 73 | \( 1 + 6.45T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 0.395T + 89T^{2} \) |
| 97 | \( 1 + 8.95T + 97T^{2} \) |
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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
−9.501049348894694417735272226134, −8.823387848317346108520743351530, −8.030575132529231181560146153612, −6.86983243172909308470023191257, −6.22923697939109420508879523543, −5.29301845460768341768850664647, −4.48825813034515604019833446902, −3.67357352188819069056700689616, −2.55746243486070783292416345739, −0.43265333212488750223838368208,
0.43265333212488750223838368208, 2.55746243486070783292416345739, 3.67357352188819069056700689616, 4.48825813034515604019833446902, 5.29301845460768341768850664647, 6.22923697939109420508879523543, 6.86983243172909308470023191257, 8.030575132529231181560146153612, 8.823387848317346108520743351530, 9.501049348894694417735272226134
