| L(s) = 1 | − 5.21·2-s + 19.2·4-s − 9.35·5-s + 26.3·7-s − 58.6·8-s + 48.8·10-s − 56.1·11-s − 11.5·13-s − 137.·14-s + 152.·16-s + 111.·17-s − 34.4·19-s − 180.·20-s + 293.·22-s + 88.9·23-s − 37.4·25-s + 60.4·26-s + 507.·28-s − 121.·29-s − 35.3·31-s − 325.·32-s − 580.·34-s − 246.·35-s − 12.9·37-s + 180.·38-s + 548.·40-s − 235.·41-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 2.40·4-s − 0.836·5-s + 1.42·7-s − 2.59·8-s + 1.54·10-s − 1.53·11-s − 0.246·13-s − 2.63·14-s + 2.38·16-s + 1.58·17-s − 0.416·19-s − 2.01·20-s + 2.84·22-s + 0.806·23-s − 0.299·25-s + 0.455·26-s + 3.42·28-s − 0.775·29-s − 0.204·31-s − 1.79·32-s − 2.92·34-s − 1.19·35-s − 0.0577·37-s + 0.768·38-s + 2.16·40-s − 0.898·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5946935050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5946935050\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 5.21T + 8T^{2} \) |
| 5 | \( 1 + 9.35T + 125T^{2} \) |
| 7 | \( 1 - 26.3T + 343T^{2} \) |
| 11 | \( 1 + 56.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 11.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 34.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 88.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 35.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 12.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 235.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 280.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 491.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 475.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 292.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 544.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 763.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 107.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 807.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 785.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 69.6T + 5.71e5T^{2} \) |
| 89 | \( 1 + 462.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 563.T + 9.12e5T^{2} \) |
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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
−8.465064301813114476826995099922, −7.911658317605536451329419143535, −7.73634588493479782375548223664, −6.94472884725677925953499732534, −5.60792389930743832689664984212, −4.92168659692097493708393312005, −3.48693182908891967151833748990, −2.41760779010653576493314082987, −1.51527521035106643332460947093, −0.47711256412682749738711108752,
0.47711256412682749738711108752, 1.51527521035106643332460947093, 2.41760779010653576493314082987, 3.48693182908891967151833748990, 4.92168659692097493708393312005, 5.60792389930743832689664984212, 6.94472884725677925953499732534, 7.73634588493479782375548223664, 7.911658317605536451329419143535, 8.465064301813114476826995099922
