| L(s) = 1 | − 3.11·3-s − 5-s + 7-s + 6.70·9-s + 1.53·11-s − 6.53·13-s + 3.11·15-s + 6.12·17-s − 4.58·19-s − 3.11·21-s + 5.80·23-s + 25-s − 11.5·27-s − 10.0·29-s − 0.568·31-s − 4.78·33-s − 35-s + 11.6·37-s + 20.3·39-s − 6.66·41-s − 43-s − 6.70·45-s − 7.52·47-s + 49-s − 19.0·51-s − 3.07·53-s − 1.53·55-s + ⋯ |
| L(s) = 1 | − 1.79·3-s − 0.447·5-s + 0.377·7-s + 2.23·9-s + 0.463·11-s − 1.81·13-s + 0.804·15-s + 1.48·17-s − 1.05·19-s − 0.679·21-s + 1.21·23-s + 0.200·25-s − 2.21·27-s − 1.85·29-s − 0.102·31-s − 0.833·33-s − 0.169·35-s + 1.91·37-s + 3.25·39-s − 1.04·41-s − 0.152·43-s − 0.998·45-s − 1.09·47-s + 0.142·49-s − 2.67·51-s − 0.422·53-s − 0.207·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 + 6.53T + 13T^{2} \) |
| 17 | \( 1 - 6.12T + 17T^{2} \) |
| 19 | \( 1 + 4.58T + 19T^{2} \) |
| 23 | \( 1 - 5.80T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 0.568T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + 6.66T + 41T^{2} \) |
| 47 | \( 1 + 7.52T + 47T^{2} \) |
| 53 | \( 1 + 3.07T + 53T^{2} \) |
| 59 | \( 1 - 4.35T + 59T^{2} \) |
| 61 | \( 1 - 3.79T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 0.391T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 7.27T + 79T^{2} \) |
| 83 | \( 1 - 4.13T + 83T^{2} \) |
| 89 | \( 1 + 9.51T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.43851081749418501540939695370, −7.04484919155056116339350179579, −6.24138623187735183232559862247, −5.46937175387423309860134303033, −4.95398059610649737180735876846, −4.38389176828725287747365055088, −3.42374690182397866385737783164, −2.07947209179749840566566308697, −0.998626947109694490440724505505, 0,
0.998626947109694490440724505505, 2.07947209179749840566566308697, 3.42374690182397866385737783164, 4.38389176828725287747365055088, 4.95398059610649737180735876846, 5.46937175387423309860134303033, 6.24138623187735183232559862247, 7.04484919155056116339350179579, 7.43851081749418501540939695370
