| L(s) = 1 | − 0.510·2-s − 3-s − 1.73·4-s − 1.60·5-s + 0.510·6-s − 0.772·7-s + 1.90·8-s + 9-s + 0.819·10-s + 5.11·11-s + 1.73·12-s − 4.25·13-s + 0.394·14-s + 1.60·15-s + 2.50·16-s − 17-s − 0.510·18-s + 0.139·19-s + 2.79·20-s + 0.772·21-s − 2.60·22-s + 2.39·23-s − 1.90·24-s − 2.41·25-s + 2.17·26-s − 27-s + 1.34·28-s + ⋯ |
| L(s) = 1 | − 0.360·2-s − 0.577·3-s − 0.869·4-s − 0.718·5-s + 0.208·6-s − 0.292·7-s + 0.674·8-s + 0.333·9-s + 0.259·10-s + 1.54·11-s + 0.502·12-s − 1.18·13-s + 0.105·14-s + 0.414·15-s + 0.626·16-s − 0.242·17-s − 0.120·18-s + 0.0320·19-s + 0.625·20-s + 0.168·21-s − 0.556·22-s + 0.500·23-s − 0.389·24-s − 0.483·25-s + 0.425·26-s − 0.192·27-s + 0.254·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6000540921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6000540921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 0.510T + 2T^{2} \) |
| 5 | \( 1 + 1.60T + 5T^{2} \) |
| 7 | \( 1 + 0.772T + 7T^{2} \) |
| 11 | \( 1 - 5.11T + 11T^{2} \) |
| 13 | \( 1 + 4.25T + 13T^{2} \) |
| 19 | \( 1 - 0.139T + 19T^{2} \) |
| 23 | \( 1 - 2.39T + 23T^{2} \) |
| 29 | \( 1 - 7.73T + 29T^{2} \) |
| 31 | \( 1 + 0.727T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 + 4.55T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 5.80T + 47T^{2} \) |
| 53 | \( 1 + 8.80T + 53T^{2} \) |
| 59 | \( 1 - 2.77T + 59T^{2} \) |
| 61 | \( 1 - 6.83T + 61T^{2} \) |
| 67 | \( 1 + 9.93T + 67T^{2} \) |
| 71 | \( 1 + 9.38T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 83 | \( 1 - 0.155T + 83T^{2} \) |
| 89 | \( 1 - 1.13T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.462867026245419591490376924808, −7.79286972044543600351268887861, −6.93198905677983721075004693783, −6.44248898496266959299986646344, −5.29668716950589453890064618981, −4.61536720995870003786531205287, −4.03447115381540188670569995698, −3.15653257406791443268559803393, −1.61086614513279614304837053161, −0.50502679549159315109542271185,
0.50502679549159315109542271185, 1.61086614513279614304837053161, 3.15653257406791443268559803393, 4.03447115381540188670569995698, 4.61536720995870003786531205287, 5.29668716950589453890064618981, 6.44248898496266959299986646344, 6.93198905677983721075004693783, 7.79286972044543600351268887861, 8.462867026245419591490376924808
