| L(s) = 1 | − 0.139·2-s + 2.98·3-s − 1.98·4-s − 0.416·6-s − 3.56·7-s + 0.554·8-s + 5.91·9-s − 5.91·12-s − 1.19·13-s + 0.497·14-s + 3.88·16-s − 5.43·17-s − 0.824·18-s + 4.80·19-s − 10.6·21-s − 1.82·23-s + 1.65·24-s + 0.166·26-s + 8.70·27-s + 7.06·28-s − 4.14·29-s + 1.28·31-s − 1.65·32-s + 0.757·34-s − 11.7·36-s − 3.16·37-s − 0.669·38-s + ⋯ |
| L(s) = 1 | − 0.0985·2-s + 1.72·3-s − 0.990·4-s − 0.169·6-s − 1.34·7-s + 0.196·8-s + 1.97·9-s − 1.70·12-s − 0.331·13-s + 0.132·14-s + 0.970·16-s − 1.31·17-s − 0.194·18-s + 1.10·19-s − 2.32·21-s − 0.379·23-s + 0.338·24-s + 0.0326·26-s + 1.67·27-s + 1.33·28-s − 0.769·29-s + 0.231·31-s − 0.291·32-s + 0.129·34-s − 1.95·36-s − 0.519·37-s − 0.108·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.139T + 2T^{2} \) |
| 3 | \( 1 - 2.98T + 3T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 13 | \( 1 + 1.19T + 13T^{2} \) |
| 17 | \( 1 + 5.43T + 17T^{2} \) |
| 19 | \( 1 - 4.80T + 19T^{2} \) |
| 23 | \( 1 + 1.82T + 23T^{2} \) |
| 29 | \( 1 + 4.14T + 29T^{2} \) |
| 31 | \( 1 - 1.28T + 31T^{2} \) |
| 37 | \( 1 + 3.16T + 37T^{2} \) |
| 41 | \( 1 + 6.40T + 41T^{2} \) |
| 43 | \( 1 - 4.23T + 43T^{2} \) |
| 47 | \( 1 + 7.02T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 4.01T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 8.14T + 67T^{2} \) |
| 71 | \( 1 - 8.51T + 71T^{2} \) |
| 73 | \( 1 + 8.54T + 73T^{2} \) |
| 79 | \( 1 - 0.393T + 79T^{2} \) |
| 83 | \( 1 + 1.56T + 83T^{2} \) |
| 89 | \( 1 + 6.90T + 89T^{2} \) |
| 97 | \( 1 + 9.31T + 97T^{2} \) |
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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
−8.518552515230393848303029490152, −7.78312679360531429993451656486, −7.09095087300768823328391874336, −6.22036372207533917207685382006, −5.01795138321759463317133393674, −4.15520288643495079677650539674, −3.41577872354401425094166202109, −2.86942004862718708317009552055, −1.68668319078212371570779885965, 0,
1.68668319078212371570779885965, 2.86942004862718708317009552055, 3.41577872354401425094166202109, 4.15520288643495079677650539674, 5.01795138321759463317133393674, 6.22036372207533917207685382006, 7.09095087300768823328391874336, 7.78312679360531429993451656486, 8.518552515230393848303029490152
