| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s − 7-s + 8-s + 9-s − 2·10-s − 12-s − 13-s − 14-s + 2·15-s + 16-s − 2·17-s + 18-s + 19-s − 2·20-s + 21-s + 23-s − 24-s − 25-s − 26-s − 27-s − 28-s + 5·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.229·19-s − 0.447·20-s + 0.218·21-s + 0.208·23-s − 0.204·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.928·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105222 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.20393244486493, −13.67720581758750, −13.01445555587525, −12.78958725291671, −12.21729173291828, −11.72459419494125, −11.46724446058876, −10.96337491877742, −10.29992065667076, −10.00941299317437, −9.291914362279314, −8.675417938536272, −8.103094709756330, −7.622273809947180, −6.967408958846992, −6.662255807396467, −6.173068836893351, −5.401028722033458, −4.913428781013119, −4.602273845463789, −3.774239353955267, −3.395325739797546, −2.891089353821754, −1.866677600541949, −1.404394547703166, 0, 0,
1.404394547703166, 1.866677600541949, 2.891089353821754, 3.395325739797546, 3.774239353955267, 4.602273845463789, 4.913428781013119, 5.401028722033458, 6.173068836893351, 6.662255807396467, 6.967408958846992, 7.622273809947180, 8.103094709756330, 8.675417938536272, 9.291914362279314, 10.00941299317437, 10.29992065667076, 10.96337491877742, 11.46724446058876, 11.72459419494125, 12.21729173291828, 12.78958725291671, 13.01445555587525, 13.67720581758750, 14.20393244486493
