| L(s) = 1 | + 0.0831·2-s − 1.99·4-s − 1.93·5-s + 0.343·7-s − 0.332·8-s − 0.160·10-s + 2.42·11-s + 3.63·13-s + 0.0285·14-s + 3.95·16-s + 6.15·17-s − 1.35·19-s + 3.84·20-s + 0.201·22-s + 3.46·23-s − 1.27·25-s + 0.301·26-s − 0.685·28-s + 10.9·31-s + 0.993·32-s + 0.511·34-s − 0.663·35-s − 3.08·37-s − 0.112·38-s + 0.640·40-s − 5.28·41-s − 0.00844·43-s + ⋯ |
| L(s) = 1 | + 0.0588·2-s − 0.996·4-s − 0.863·5-s + 0.129·7-s − 0.117·8-s − 0.0507·10-s + 0.730·11-s + 1.00·13-s + 0.00764·14-s + 0.989·16-s + 1.49·17-s − 0.309·19-s + 0.860·20-s + 0.0429·22-s + 0.722·23-s − 0.254·25-s + 0.0592·26-s − 0.129·28-s + 1.97·31-s + 0.175·32-s + 0.0877·34-s − 0.112·35-s − 0.507·37-s − 0.0182·38-s + 0.101·40-s − 0.825·41-s − 0.00128·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.557859621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.557859621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 0.0831T + 2T^{2} \) |
| 5 | \( 1 + 1.93T + 5T^{2} \) |
| 7 | \( 1 - 0.343T + 7T^{2} \) |
| 11 | \( 1 - 2.42T + 11T^{2} \) |
| 13 | \( 1 - 3.63T + 13T^{2} \) |
| 17 | \( 1 - 6.15T + 17T^{2} \) |
| 19 | \( 1 + 1.35T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 + 3.08T + 37T^{2} \) |
| 41 | \( 1 + 5.28T + 41T^{2} \) |
| 43 | \( 1 + 0.00844T + 43T^{2} \) |
| 47 | \( 1 - 5.79T + 47T^{2} \) |
| 53 | \( 1 - 4.14T + 53T^{2} \) |
| 59 | \( 1 + 5.76T + 59T^{2} \) |
| 61 | \( 1 + 6.23T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 0.459T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 2.87T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 2.72T + 97T^{2} \) |
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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
−7.975290220886155502592716458771, −7.37351196471227834058832651596, −6.39221520778612307925936373271, −5.78741643664657817163740311876, −4.89701630995171847474963232563, −4.29335639596900699488492095342, −3.58731462954538532817590145292, −3.09551287850145345631067777469, −1.45888059965780656837024871940, −0.69538489021728697424033390715,
0.69538489021728697424033390715, 1.45888059965780656837024871940, 3.09551287850145345631067777469, 3.58731462954538532817590145292, 4.29335639596900699488492095342, 4.89701630995171847474963232563, 5.78741643664657817163740311876, 6.39221520778612307925936373271, 7.37351196471227834058832651596, 7.975290220886155502592716458771
