| L(s) = 1 | + 1.80·2-s + 1.24·4-s − 0.356·5-s − 4.04·7-s − 1.35·8-s − 0.643·10-s − 2.91·11-s + 5.18·13-s − 7.29·14-s − 4.93·16-s − 1.10·17-s + 2.04·19-s − 0.445·20-s − 5.24·22-s + 4.13·23-s − 4.87·25-s + 9.34·26-s − 5.04·28-s − 6.35·31-s − 6.18·32-s − 2·34-s + 1.44·35-s − 2.91·37-s + 3.69·38-s + 0.484·40-s − 0.396·41-s + 5.74·43-s + ⋯ |
| L(s) = 1 | + 1.27·2-s + 0.623·4-s − 0.159·5-s − 1.53·7-s − 0.479·8-s − 0.203·10-s − 0.877·11-s + 1.43·13-s − 1.94·14-s − 1.23·16-s − 0.269·17-s + 0.470·19-s − 0.0995·20-s − 1.11·22-s + 0.862·23-s − 0.974·25-s + 1.83·26-s − 0.954·28-s − 1.14·31-s − 1.09·32-s − 0.342·34-s + 0.244·35-s − 0.478·37-s + 0.598·38-s + 0.0765·40-s − 0.0618·41-s + 0.875·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.271336757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.271336757\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 5 | \( 1 + 0.356T + 5T^{2} \) |
| 7 | \( 1 + 4.04T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 - 5.18T + 13T^{2} \) |
| 17 | \( 1 + 1.10T + 17T^{2} \) |
| 19 | \( 1 - 2.04T + 19T^{2} \) |
| 23 | \( 1 - 4.13T + 23T^{2} \) |
| 31 | \( 1 + 6.35T + 31T^{2} \) |
| 37 | \( 1 + 2.91T + 37T^{2} \) |
| 41 | \( 1 + 0.396T + 41T^{2} \) |
| 43 | \( 1 - 5.74T + 43T^{2} \) |
| 47 | \( 1 - 7.80T + 47T^{2} \) |
| 53 | \( 1 - 4.35T + 53T^{2} \) |
| 59 | \( 1 - 9.10T + 59T^{2} \) |
| 61 | \( 1 + 6.04T + 61T^{2} \) |
| 67 | \( 1 - 0.374T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 8.94T + 73T^{2} \) |
| 79 | \( 1 + 0.594T + 79T^{2} \) |
| 83 | \( 1 - 9.43T + 83T^{2} \) |
| 89 | \( 1 + 1.42T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−7.62580573909976063010793687368, −6.96776273860217109036718928445, −6.22802608963871781689376516394, −5.75699447846335466780540290003, −5.17045972930910578714647514279, −4.12418571900202121852627555386, −3.58259048879244048283631906440, −3.08614372714616339844536472289, −2.20206065351234175941156685553, −0.59129113222224349687201927307,
0.59129113222224349687201927307, 2.20206065351234175941156685553, 3.08614372714616339844536472289, 3.58259048879244048283631906440, 4.12418571900202121852627555386, 5.17045972930910578714647514279, 5.75699447846335466780540290003, 6.22802608963871781689376516394, 6.96776273860217109036718928445, 7.62580573909976063010793687368
