| L(s) = 1 | + 2-s + 4-s + 0.586·5-s − 1.49·7-s + 8-s + 0.586·10-s + 2.43·11-s + 3.04·13-s − 1.49·14-s + 16-s + 3.77·17-s + 5.03·19-s + 0.586·20-s + 2.43·22-s − 3.85·23-s − 4.65·25-s + 3.04·26-s − 1.49·28-s − 6.94·29-s + 0.584·31-s + 32-s + 3.77·34-s − 0.879·35-s + 4.55·37-s + 5.03·38-s + 0.586·40-s + 11.2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.262·5-s − 0.566·7-s + 0.353·8-s + 0.185·10-s + 0.735·11-s + 0.844·13-s − 0.400·14-s + 0.250·16-s + 0.914·17-s + 1.15·19-s + 0.131·20-s + 0.519·22-s − 0.802·23-s − 0.931·25-s + 0.596·26-s − 0.283·28-s − 1.28·29-s + 0.105·31-s + 0.176·32-s + 0.646·34-s − 0.148·35-s + 0.748·37-s + 0.816·38-s + 0.0927·40-s + 1.75·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.706524916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.706524916\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 353 | \( 1 - T \) |
| good | 5 | \( 1 - 0.586T + 5T^{2} \) |
| 7 | \( 1 + 1.49T + 7T^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 13 | \( 1 - 3.04T + 13T^{2} \) |
| 17 | \( 1 - 3.77T + 17T^{2} \) |
| 19 | \( 1 - 5.03T + 19T^{2} \) |
| 23 | \( 1 + 3.85T + 23T^{2} \) |
| 29 | \( 1 + 6.94T + 29T^{2} \) |
| 31 | \( 1 - 0.584T + 31T^{2} \) |
| 37 | \( 1 - 4.55T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 0.866T + 43T^{2} \) |
| 47 | \( 1 - 7.16T + 47T^{2} \) |
| 53 | \( 1 - 7.64T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 1.98T + 67T^{2} \) |
| 71 | \( 1 + 6.42T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 0.203T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 - 5.55T + 89T^{2} \) |
| 97 | \( 1 - 2.23T + 97T^{2} \) |
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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
−7.66270857559910367662232002618, −7.45414508159603487892650552371, −6.25902244255898722231394344507, −5.98139479980515336711514635066, −5.35504126196070579638417730046, −4.19185015610739744385020450243, −3.71684734412190158695874120410, −2.97080673007882257122556551258, −1.91331127702490677987081028407, −0.940124499943691839917109158994,
0.940124499943691839917109158994, 1.91331127702490677987081028407, 2.97080673007882257122556551258, 3.71684734412190158695874120410, 4.19185015610739744385020450243, 5.35504126196070579638417730046, 5.98139479980515336711514635066, 6.25902244255898722231394344507, 7.45414508159603487892650552371, 7.66270857559910367662232002618
