| L(s) = 1 | + 1.48·2-s + 0.193·4-s + 4.48·7-s − 2.67·8-s − 3.67·11-s + 2.86·13-s + 6.63·14-s − 4.35·16-s − 6.15·17-s − 19-s − 5.44·22-s − 8.15·23-s + 4.24·26-s + 0.869·28-s − 4.63·29-s − 2.80·31-s − 1.09·32-s − 9.11·34-s − 3.44·37-s − 1.48·38-s + 5.59·41-s − 10.1·43-s − 0.712·44-s − 12.0·46-s − 3.84·47-s + 13.0·49-s + 0.556·52-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.0969·4-s + 1.69·7-s − 0.945·8-s − 1.10·11-s + 0.795·13-s + 1.77·14-s − 1.08·16-s − 1.49·17-s − 0.229·19-s − 1.16·22-s − 1.70·23-s + 0.833·26-s + 0.164·28-s − 0.861·29-s − 0.503·31-s − 0.193·32-s − 1.56·34-s − 0.566·37-s − 0.240·38-s + 0.874·41-s − 1.54·43-s − 0.107·44-s − 1.78·46-s − 0.560·47-s + 1.86·49-s + 0.0771·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 7 | \( 1 - 4.48T + 7T^{2} \) |
| 11 | \( 1 + 3.67T + 11T^{2} \) |
| 13 | \( 1 - 2.86T + 13T^{2} \) |
| 17 | \( 1 + 6.15T + 17T^{2} \) |
| 23 | \( 1 + 8.15T + 23T^{2} \) |
| 29 | \( 1 + 4.63T + 29T^{2} \) |
| 31 | \( 1 + 2.80T + 31T^{2} \) |
| 37 | \( 1 + 3.44T + 37T^{2} \) |
| 41 | \( 1 - 5.59T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 3.84T + 47T^{2} \) |
| 53 | \( 1 - 3.89T + 53T^{2} \) |
| 59 | \( 1 - 9.53T + 59T^{2} \) |
| 61 | \( 1 + 9.89T + 61T^{2} \) |
| 67 | \( 1 - 8.70T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 3.08T + 73T^{2} \) |
| 79 | \( 1 - 4.96T + 79T^{2} \) |
| 83 | \( 1 + 6.73T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 9.63T + 97T^{2} \) |
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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
−8.254060194464370458530843892728, −7.26042420277940743411015818509, −6.29464459542632045769713364698, −5.54851664747922271800707779699, −5.00855040844894080549152045214, −4.29779687760289000978172683035, −3.71983181297957502588035087983, −2.44789917193775925686595693432, −1.77721469464315417380654339946, 0,
1.77721469464315417380654339946, 2.44789917193775925686595693432, 3.71983181297957502588035087983, 4.29779687760289000978172683035, 5.00855040844894080549152045214, 5.54851664747922271800707779699, 6.29464459542632045769713364698, 7.26042420277940743411015818509, 8.254060194464370458530843892728
