| L(s) = 1 | − 0.308·2-s − 2.95·3-s − 1.90·4-s − 3.87·5-s + 0.912·6-s − 7-s + 1.20·8-s + 5.72·9-s + 1.19·10-s − 2.74·11-s + 5.62·12-s − 13-s + 0.308·14-s + 11.4·15-s + 3.43·16-s − 17-s − 1.76·18-s − 0.279·19-s + 7.37·20-s + 2.95·21-s + 0.846·22-s + 5.09·23-s − 3.56·24-s + 10.0·25-s + 0.308·26-s − 8.06·27-s + 1.90·28-s + ⋯ |
| L(s) = 1 | − 0.218·2-s − 1.70·3-s − 0.952·4-s − 1.73·5-s + 0.372·6-s − 0.377·7-s + 0.426·8-s + 1.90·9-s + 0.378·10-s − 0.826·11-s + 1.62·12-s − 0.277·13-s + 0.0825·14-s + 2.95·15-s + 0.859·16-s − 0.242·17-s − 0.416·18-s − 0.0642·19-s + 1.65·20-s + 0.644·21-s + 0.180·22-s + 1.06·23-s − 0.726·24-s + 2.00·25-s + 0.0605·26-s − 1.55·27-s + 0.359·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 0.308T + 2T^{2} \) |
| 3 | \( 1 + 2.95T + 3T^{2} \) |
| 5 | \( 1 + 3.87T + 5T^{2} \) |
| 11 | \( 1 + 2.74T + 11T^{2} \) |
| 19 | \( 1 + 0.279T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 - 8.61T + 29T^{2} \) |
| 31 | \( 1 - 0.289T + 31T^{2} \) |
| 37 | \( 1 + 9.41T + 37T^{2} \) |
| 41 | \( 1 + 6.11T + 41T^{2} \) |
| 43 | \( 1 - 1.71T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 0.737T + 53T^{2} \) |
| 59 | \( 1 + 5.45T + 59T^{2} \) |
| 61 | \( 1 - 7.03T + 61T^{2} \) |
| 67 | \( 1 - 0.939T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 + 5.45T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 5.00T + 89T^{2} \) |
| 97 | \( 1 + 0.961T + 97T^{2} \) |
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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
−8.948820542857137253243446303448, −8.233059794539154057993503315536, −7.31988771037947630637213355299, −6.77307182042551117905231874881, −5.52602946444443797451199764682, −4.83385495535320922363563992846, −4.29357928319863582455100568633, −3.22561922506862898459818590706, −0.829778937782121185916278972359, 0,
0.829778937782121185916278972359, 3.22561922506862898459818590706, 4.29357928319863582455100568633, 4.83385495535320922363563992846, 5.52602946444443797451199764682, 6.77307182042551117905231874881, 7.31988771037947630637213355299, 8.233059794539154057993503315536, 8.948820542857137253243446303448
