| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 2·11-s + 6·13-s − 14-s + 16-s + 6·17-s + 19-s + 2·22-s + 3·23-s − 6·26-s + 28-s + 2·29-s − 10·31-s − 32-s − 6·34-s + 2·37-s − 38-s − 9·41-s + 8·43-s − 2·44-s − 3·46-s + 8·47-s − 6·49-s + 6·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.603·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.229·19-s + 0.426·22-s + 0.625·23-s − 1.17·26-s + 0.188·28-s + 0.371·29-s − 1.79·31-s − 0.176·32-s − 1.02·34-s + 0.328·37-s − 0.162·38-s − 1.40·41-s + 1.21·43-s − 0.301·44-s − 0.442·46-s + 1.16·47-s − 6/7·49-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.277665699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.277665699\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.15596823036006, −13.46763926859860, −13.08087878475913, −12.60710993683609, −11.77958633219525, −11.63427691318255, −10.91619694863408, −10.48595475893873, −10.21225173381807, −9.439499214431272, −8.874039473266259, −8.567546781770708, −7.992577420799803, −7.380983822844583, −7.144914371019751, −6.250134882570140, −5.660998476434981, −5.457175220632354, −4.589669002470346, −3.742868711633417, −3.374845947770883, −2.662892320318091, −1.824061341622156, −1.218101253019796, −0.6207880918755594,
0.6207880918755594, 1.218101253019796, 1.824061341622156, 2.662892320318091, 3.374845947770883, 3.742868711633417, 4.589669002470346, 5.457175220632354, 5.660998476434981, 6.250134882570140, 7.144914371019751, 7.380983822844583, 7.992577420799803, 8.567546781770708, 8.874039473266259, 9.439499214431272, 10.21225173381807, 10.48595475893873, 10.91619694863408, 11.63427691318255, 11.77958633219525, 12.60710993683609, 13.08087878475913, 13.46763926859860, 14.15596823036006
