| L(s) = 1 | − 2-s + 2.64·3-s + 4-s − 1.64·5-s − 2.64·6-s − 8-s + 4.00·9-s + 1.64·10-s + 11-s + 2.64·12-s + 5·13-s − 4.35·15-s + 16-s + 6·17-s − 4.00·18-s − 5.64·19-s − 1.64·20-s − 22-s + 1.64·23-s − 2.64·24-s − 2.29·25-s − 5·26-s + 2.64·27-s + 6.29·29-s + 4.35·30-s − 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.52·3-s + 0.5·4-s − 0.736·5-s − 1.08·6-s − 0.353·8-s + 1.33·9-s + 0.520·10-s + 0.301·11-s + 0.763·12-s + 1.38·13-s − 1.12·15-s + 0.250·16-s + 1.45·17-s − 0.942·18-s − 1.29·19-s − 0.368·20-s − 0.213·22-s + 0.343·23-s − 0.540·24-s − 0.458·25-s − 0.980·26-s + 0.509·27-s + 1.16·29-s + 0.794·30-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.911402955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.911402955\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2.64T + 3T^{2} \) |
| 5 | \( 1 + 1.64T + 5T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 5.64T + 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 - 6.29T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 3.64T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 2.70T + 47T^{2} \) |
| 53 | \( 1 - 1.64T + 53T^{2} \) |
| 59 | \( 1 - 4.64T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 4.35T + 71T^{2} \) |
| 73 | \( 1 - 0.354T + 73T^{2} \) |
| 79 | \( 1 + 2.64T + 79T^{2} \) |
| 83 | \( 1 - 2.70T + 83T^{2} \) |
| 89 | \( 1 - 6.58T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−9.632537028798686705157408053449, −8.870067119571187852377744132953, −8.257374821057161469201574906188, −7.82901850870273619132363586178, −6.87824828145069999221656695653, −5.83362987430832418518749912151, −4.12640835014932227321170940254, −3.53926021521704570630852993250, −2.50891854749754943811744453494, −1.20397846388702966336886662298,
1.20397846388702966336886662298, 2.50891854749754943811744453494, 3.53926021521704570630852993250, 4.12640835014932227321170940254, 5.83362987430832418518749912151, 6.87824828145069999221656695653, 7.82901850870273619132363586178, 8.257374821057161469201574906188, 8.870067119571187852377744132953, 9.632537028798686705157408053449
