L(s) = 1 | − 1.26·3-s + 3.32·5-s + 2.61·7-s − 1.39·9-s + 5.93·11-s − 5.46·13-s − 4.20·15-s + 1.40·17-s + 0.119·19-s − 3.30·21-s + 2.06·23-s + 6.04·25-s + 5.56·27-s + 7.76·29-s − 3.19·31-s − 7.50·33-s + 8.68·35-s − 10.7·37-s + 6.92·39-s − 3.99·41-s + 4.99·43-s − 4.64·45-s + 9.89·47-s − 0.166·49-s − 1.77·51-s + 1.77·53-s + 19.7·55-s + ⋯ |
L(s) = 1 | − 0.730·3-s + 1.48·5-s + 0.988·7-s − 0.465·9-s + 1.78·11-s − 1.51·13-s − 1.08·15-s + 0.340·17-s + 0.0274·19-s − 0.722·21-s + 0.431·23-s + 1.20·25-s + 1.07·27-s + 1.44·29-s − 0.573·31-s − 1.30·33-s + 1.46·35-s − 1.76·37-s + 1.10·39-s − 0.624·41-s + 0.761·43-s − 0.692·45-s + 1.44·47-s − 0.0237·49-s − 0.248·51-s + 0.243·53-s + 2.65·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.600296772\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.600296772\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 547 | \( 1 + T \) |
good | 3 | \( 1 + 1.26T + 3T^{2} \) |
| 5 | \( 1 - 3.32T + 5T^{2} \) |
| 7 | \( 1 - 2.61T + 7T^{2} \) |
| 11 | \( 1 - 5.93T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 - 0.119T + 19T^{2} \) |
| 23 | \( 1 - 2.06T + 23T^{2} \) |
| 29 | \( 1 - 7.76T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 3.99T + 41T^{2} \) |
| 43 | \( 1 - 4.99T + 43T^{2} \) |
| 47 | \( 1 - 9.89T + 47T^{2} \) |
| 53 | \( 1 - 1.77T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 0.508T + 61T^{2} \) |
| 67 | \( 1 - 0.992T + 67T^{2} \) |
| 71 | \( 1 + 4.90T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 8.27T + 89T^{2} \) |
| 97 | \( 1 - 2.09T + 97T^{2} \) |
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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
−7.63436314756279546722258003204, −6.82883475273825563146642033868, −6.37359884897592034967952197215, −5.65846707172528346093808193405, −5.03544186274338558971899053778, −4.64059911513287121636447267249, −3.43127390249585191651524869830, −2.39424459485740641897835941858, −1.70845959632337424369788160811, −0.857079051578635774485994362801,
0.857079051578635774485994362801, 1.70845959632337424369788160811, 2.39424459485740641897835941858, 3.43127390249585191651524869830, 4.64059911513287121636447267249, 5.03544186274338558971899053778, 5.65846707172528346093808193405, 6.37359884897592034967952197215, 6.82883475273825563146642033868, 7.63436314756279546722258003204