| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 11-s − 2·13-s + 16-s + 2·17-s − 20-s − 22-s + 4·23-s + 25-s + 2·26-s + 6·29-s − 8·31-s − 32-s − 2·34-s − 6·37-s + 40-s + 2·41-s + 4·43-s + 44-s − 4·46-s − 12·47-s − 7·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s + 1.11·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s − 0.986·37-s + 0.158·40-s + 0.312·41-s + 0.609·43-s + 0.150·44-s − 0.589·46-s − 1.75·47-s − 49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4169020733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4169020733\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.47952782900105, −12.08213264076781, −11.46756371189983, −11.13413473038023, −10.84619322038300, −10.09167087777424, −9.853621871731617, −9.357226162106292, −8.900905482781920, −8.343958205904495, −8.084412181764108, −7.446375426712598, −7.100628348141456, −6.640092455909387, −6.194880356094829, −5.470818101728364, −5.010693156842358, −4.629367895682077, −3.735880502516569, −3.499859463986264, −2.794222316379262, −2.347180703898031, −1.455849343699882, −1.201702076188083, −0.1961131015230510,
0.1961131015230510, 1.201702076188083, 1.455849343699882, 2.347180703898031, 2.794222316379262, 3.499859463986264, 3.735880502516569, 4.629367895682077, 5.010693156842358, 5.470818101728364, 6.194880356094829, 6.640092455909387, 7.100628348141456, 7.446375426712598, 8.084412181764108, 8.343958205904495, 8.900905482781920, 9.357226162106292, 9.853621871731617, 10.09167087777424, 10.84619322038300, 11.13413473038023, 11.46756371189983, 12.08213264076781, 12.47952782900105
