| L(s) = 1 | − 2.53·2-s + 3-s + 4.42·4-s − 2.65·5-s − 2.53·6-s + 1.73·7-s − 6.14·8-s + 9-s + 6.73·10-s − 1.76·11-s + 4.42·12-s − 2.94·13-s − 4.39·14-s − 2.65·15-s + 6.72·16-s + 17-s − 2.53·18-s + 1.53·19-s − 11.7·20-s + 1.73·21-s + 4.48·22-s − 1.28·23-s − 6.14·24-s + 2.05·25-s + 7.47·26-s + 27-s + 7.66·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.577·3-s + 2.21·4-s − 1.18·5-s − 1.03·6-s + 0.654·7-s − 2.17·8-s + 0.333·9-s + 2.12·10-s − 0.533·11-s + 1.27·12-s − 0.818·13-s − 1.17·14-s − 0.685·15-s + 1.68·16-s + 0.242·17-s − 0.597·18-s + 0.351·19-s − 2.62·20-s + 0.377·21-s + 0.956·22-s − 0.268·23-s − 1.25·24-s + 0.410·25-s + 1.46·26-s + 0.192·27-s + 1.44·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 5 | \( 1 + 2.65T + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 + 1.76T + 11T^{2} \) |
| 13 | \( 1 + 2.94T + 13T^{2} \) |
| 19 | \( 1 - 1.53T + 19T^{2} \) |
| 23 | \( 1 + 1.28T + 23T^{2} \) |
| 29 | \( 1 + 3.07T + 29T^{2} \) |
| 31 | \( 1 - 7.59T + 31T^{2} \) |
| 37 | \( 1 - 3.88T + 37T^{2} \) |
| 41 | \( 1 - 5.59T + 41T^{2} \) |
| 43 | \( 1 - 1.32T + 43T^{2} \) |
| 47 | \( 1 + 9.39T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 3.88T + 61T^{2} \) |
| 67 | \( 1 - 7.32T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 7.38T + 73T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 + 1.23T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−8.155768039026589471506297887564, −7.60599208223272699776530567224, −7.26226786904346810689684374843, −6.25081552801103624424911116893, −5.02753089945962414479490909796, −4.12760942480014449627921799010, −3.01164765189869274715644701811, −2.27253959799165872193181171201, −1.15677276347120226328498314200, 0,
1.15677276347120226328498314200, 2.27253959799165872193181171201, 3.01164765189869274715644701811, 4.12760942480014449627921799010, 5.02753089945962414479490909796, 6.25081552801103624424911116893, 7.26226786904346810689684374843, 7.60599208223272699776530567224, 8.155768039026589471506297887564
