L(s) = 1 | + 2-s − 3-s + 4-s + 2.30·5-s − 6-s − 1.48·7-s + 8-s + 9-s + 2.30·10-s + 1.45·11-s − 12-s − 13-s − 1.48·14-s − 2.30·15-s + 16-s − 5.05·17-s + 18-s + 1.16·19-s + 2.30·20-s + 1.48·21-s + 1.45·22-s + 1.42·23-s − 24-s + 0.299·25-s − 26-s − 27-s − 1.48·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.02·5-s − 0.408·6-s − 0.560·7-s + 0.353·8-s + 0.333·9-s + 0.727·10-s + 0.438·11-s − 0.288·12-s − 0.277·13-s − 0.396·14-s − 0.594·15-s + 0.250·16-s − 1.22·17-s + 0.235·18-s + 0.266·19-s + 0.514·20-s + 0.323·21-s + 0.310·22-s + 0.296·23-s − 0.204·24-s + 0.0598·25-s − 0.196·26-s − 0.192·27-s − 0.280·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.033171218\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.033171218\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 5 | \( 1 - 2.30T + 5T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 - 1.16T + 19T^{2} \) |
| 23 | \( 1 - 1.42T + 23T^{2} \) |
| 29 | \( 1 + 0.645T + 29T^{2} \) |
| 31 | \( 1 - 5.49T + 31T^{2} \) |
| 37 | \( 1 - 9.13T + 37T^{2} \) |
| 41 | \( 1 + 3.13T + 41T^{2} \) |
| 43 | \( 1 - 9.55T + 43T^{2} \) |
| 47 | \( 1 + 0.649T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 4.11T + 59T^{2} \) |
| 61 | \( 1 - 6.41T + 61T^{2} \) |
| 67 | \( 1 + 8.73T + 67T^{2} \) |
| 71 | \( 1 - 5.64T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 83 | \( 1 - 9.50T + 83T^{2} \) |
| 89 | \( 1 - 2.12T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−7.86151408715933051153075041414, −7.02254809700826249654840407000, −6.36137101753623750857172868304, −6.06480480710463994554351082946, −5.22095197285626908439434741034, −4.54631073797088580372867362463, −3.77716294236854252703020761519, −2.68629112278556564879061809494, −2.04601444428080893205698272345, −0.844371341722138636244060348734,
0.844371341722138636244060348734, 2.04601444428080893205698272345, 2.68629112278556564879061809494, 3.77716294236854252703020761519, 4.54631073797088580372867362463, 5.22095197285626908439434741034, 6.06480480710463994554351082946, 6.36137101753623750857172868304, 7.02254809700826249654840407000, 7.86151408715933051153075041414