| L(s) = 1 | + 2-s − 3-s + 4-s + 4.27·5-s − 6-s − 7-s + 8-s + 9-s + 4.27·10-s + 2.27·11-s − 12-s − 14-s − 4.27·15-s + 16-s + 0.274·17-s + 18-s + 2.27·19-s + 4.27·20-s + 21-s + 2.27·22-s + 2.27·23-s − 24-s + 13.2·25-s − 27-s − 28-s + 8.27·29-s − 4.27·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.91·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.35·10-s + 0.685·11-s − 0.288·12-s − 0.267·14-s − 1.10·15-s + 0.250·16-s + 0.0666·17-s + 0.235·18-s + 0.521·19-s + 0.955·20-s + 0.218·21-s + 0.485·22-s + 0.474·23-s − 0.204·24-s + 2.65·25-s − 0.192·27-s − 0.188·28-s + 1.53·29-s − 0.780·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.351017379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.351017379\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 4.27T + 5T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 17 | \( 1 - 0.274T + 17T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 23 | \( 1 - 2.27T + 23T^{2} \) |
| 29 | \( 1 - 8.27T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 4.27T + 37T^{2} \) |
| 41 | \( 1 + 6.54T + 41T^{2} \) |
| 43 | \( 1 + 2.27T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 4.54T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−7.68634971261731841361530345503, −6.75219427046077400445373489633, −6.45392935700676377918815887291, −5.82210436202654263168435037062, −5.20828170826158105598078173925, −4.64435039634744777431865735220, −3.51738931187456474759756935512, −2.72574481248970376630769618616, −1.82285674105995606144252322200, −1.05970542514761891794884568006,
1.05970542514761891794884568006, 1.82285674105995606144252322200, 2.72574481248970376630769618616, 3.51738931187456474759756935512, 4.64435039634744777431865735220, 5.20828170826158105598078173925, 5.82210436202654263168435037062, 6.45392935700676377918815887291, 6.75219427046077400445373489633, 7.68634971261731841361530345503
