| L(s) = 1 | − 0.717·2-s − 0.150·3-s − 1.48·4-s + 0.107·6-s + 4.77·7-s + 2.50·8-s − 2.97·9-s + 5.46·11-s + 0.223·12-s + 2.33·13-s − 3.42·14-s + 1.17·16-s − 3.50·17-s + 2.13·18-s + 2.54·19-s − 0.717·21-s − 3.91·22-s − 2.57·23-s − 0.375·24-s − 1.67·26-s + 0.897·27-s − 7.09·28-s + 5.43·29-s − 3.03·31-s − 5.84·32-s − 0.820·33-s + 2.51·34-s + ⋯ |
| L(s) = 1 | − 0.507·2-s − 0.0866·3-s − 0.742·4-s + 0.0439·6-s + 1.80·7-s + 0.883·8-s − 0.992·9-s + 1.64·11-s + 0.0643·12-s + 0.647·13-s − 0.916·14-s + 0.294·16-s − 0.851·17-s + 0.503·18-s + 0.582·19-s − 0.156·21-s − 0.835·22-s − 0.536·23-s − 0.0766·24-s − 0.328·26-s + 0.172·27-s − 1.34·28-s + 1.00·29-s − 0.545·31-s − 1.03·32-s − 0.142·33-s + 0.431·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.356543354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.356543354\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.717T + 2T^{2} \) |
| 3 | \( 1 + 0.150T + 3T^{2} \) |
| 7 | \( 1 - 4.77T + 7T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 - 2.33T + 13T^{2} \) |
| 17 | \( 1 + 3.50T + 17T^{2} \) |
| 19 | \( 1 - 2.54T + 19T^{2} \) |
| 23 | \( 1 + 2.57T + 23T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 + 3.03T + 31T^{2} \) |
| 37 | \( 1 + 4.78T + 37T^{2} \) |
| 41 | \( 1 - 3.59T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 1.78T + 59T^{2} \) |
| 67 | \( 1 + 6.87T + 67T^{2} \) |
| 71 | \( 1 + 8.67T + 71T^{2} \) |
| 73 | \( 1 + 3.61T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 2.60T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−8.985419841354099338444571469883, −8.839230019569748549886767913443, −8.156003390260830831774854055009, −7.26897189797600693230037697328, −6.15730716707852290489005962983, −5.21998158052433410485016036387, −4.46361925621617159406337912953, −3.66059253760733198450666131416, −1.93425294836652749110637227331, −0.980365616948382417723837153742,
0.980365616948382417723837153742, 1.93425294836652749110637227331, 3.66059253760733198450666131416, 4.46361925621617159406337912953, 5.21998158052433410485016036387, 6.15730716707852290489005962983, 7.26897189797600693230037697328, 8.156003390260830831774854055009, 8.839230019569748549886767913443, 8.985419841354099338444571469883
