| L(s) = 1 | + 3-s − 1.04·5-s − 0.554·7-s + 9-s − 2.91·11-s − 1.04·15-s − 4.85·17-s + 0.753·19-s − 0.554·21-s − 5.76·23-s − 3.89·25-s + 27-s − 1.91·29-s + 9.51·31-s − 2.91·33-s + 0.582·35-s + 5.75·37-s + 4.91·41-s + 11.0·43-s − 1.04·45-s − 0.753·47-s − 6.69·49-s − 4.85·51-s − 7.58·53-s + 3.05·55-s + 0.753·57-s − 4.09·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.469·5-s − 0.209·7-s + 0.333·9-s − 0.877·11-s − 0.270·15-s − 1.17·17-s + 0.172·19-s − 0.121·21-s − 1.20·23-s − 0.779·25-s + 0.192·27-s − 0.355·29-s + 1.70·31-s − 0.506·33-s + 0.0983·35-s + 0.945·37-s + 0.767·41-s + 1.69·43-s − 0.156·45-s − 0.109·47-s − 0.956·49-s − 0.679·51-s − 1.04·53-s + 0.411·55-s + 0.0997·57-s − 0.533·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.606488960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.606488960\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 1.04T + 5T^{2} \) |
| 7 | \( 1 + 0.554T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 19 | \( 1 - 0.753T + 19T^{2} \) |
| 23 | \( 1 + 5.76T + 23T^{2} \) |
| 29 | \( 1 + 1.91T + 29T^{2} \) |
| 31 | \( 1 - 9.51T + 31T^{2} \) |
| 37 | \( 1 - 5.75T + 37T^{2} \) |
| 41 | \( 1 - 4.91T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 0.753T + 47T^{2} \) |
| 53 | \( 1 + 7.58T + 53T^{2} \) |
| 59 | \( 1 + 4.09T + 59T^{2} \) |
| 61 | \( 1 + 3.42T + 61T^{2} \) |
| 67 | \( 1 - 1.87T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 1.33T + 79T^{2} \) |
| 83 | \( 1 + 2.64T + 83T^{2} \) |
| 89 | \( 1 - 9.92T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−7.989040461618298438918402838153, −7.34220319465195336011844332495, −6.38443855629233597173092008936, −5.92358240890564138096217082712, −4.77660750054017168967088323643, −4.30033267057711626040557373489, −3.49011179170776805892499418437, −2.61660583421885651687757797999, −2.01918009743368334606258173453, −0.58071451581719684854969241450,
0.58071451581719684854969241450, 2.01918009743368334606258173453, 2.61660583421885651687757797999, 3.49011179170776805892499418437, 4.30033267057711626040557373489, 4.77660750054017168967088323643, 5.92358240890564138096217082712, 6.38443855629233597173092008936, 7.34220319465195336011844332495, 7.989040461618298438918402838153
