| L(s) = 1 | + 1.07·2-s + 3-s − 0.854·4-s + 1.07·6-s − 0.661·7-s − 3.05·8-s + 9-s − 0.854·12-s + 4.53·13-s − 0.708·14-s − 1.56·16-s − 3.87·17-s + 1.07·18-s + 1.73·19-s − 0.661·21-s + 7.85·23-s − 3.05·24-s + 4.85·26-s + 27-s + 0.565·28-s − 0.408·29-s − 0.145·31-s + 4.43·32-s − 4.14·34-s − 0.854·36-s − 7.70·37-s + 1.85·38-s + ⋯ |
| L(s) = 1 | + 0.756·2-s + 0.577·3-s − 0.427·4-s + 0.437·6-s − 0.250·7-s − 1.08·8-s + 0.333·9-s − 0.246·12-s + 1.25·13-s − 0.189·14-s − 0.390·16-s − 0.939·17-s + 0.252·18-s + 0.397·19-s − 0.144·21-s + 1.63·23-s − 0.623·24-s + 0.951·26-s + 0.192·27-s + 0.106·28-s − 0.0759·29-s − 0.0262·31-s + 0.784·32-s − 0.711·34-s − 0.142·36-s − 1.26·37-s + 0.300·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.040059066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.040059066\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.07T + 2T^{2} \) |
| 7 | \( 1 + 0.661T + 7T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 - 7.85T + 23T^{2} \) |
| 29 | \( 1 + 0.408T + 29T^{2} \) |
| 31 | \( 1 + 0.145T + 31T^{2} \) |
| 37 | \( 1 + 7.70T + 37T^{2} \) |
| 41 | \( 1 + 0.661T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 3.70T + 47T^{2} \) |
| 53 | \( 1 + 4.14T + 53T^{2} \) |
| 59 | \( 1 - 6.70T + 59T^{2} \) |
| 61 | \( 1 + 7.74T + 61T^{2} \) |
| 67 | \( 1 - 14T + 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 2.39T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 + 6.67T + 83T^{2} \) |
| 89 | \( 1 - 8.56T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−7.80269322321744550886555498623, −6.76318460635432511360588850043, −6.47429670635666755603752918358, −5.46105971388036662746234658285, −4.94179575476123166094438737905, −4.14005833828709268158995907392, −3.44881934174686823319884521661, −3.01439594935525509534931420217, −1.89460117692904593710781594397, −0.73547589046576370981107792273,
0.73547589046576370981107792273, 1.89460117692904593710781594397, 3.01439594935525509534931420217, 3.44881934174686823319884521661, 4.14005833828709268158995907392, 4.94179575476123166094438737905, 5.46105971388036662746234658285, 6.47429670635666755603752918358, 6.76318460635432511360588850043, 7.80269322321744550886555498623
