| L(s) = 1 | − 2.36·2-s − 3-s + 3.58·4-s + 2.36·6-s + 1.44·7-s − 3.74·8-s + 9-s + 4.67·11-s − 3.58·12-s + 0.873·13-s − 3.42·14-s + 1.67·16-s − 2.24·17-s − 2.36·18-s − 5.64·19-s − 1.44·21-s − 11.0·22-s − 5.37·23-s + 3.74·24-s − 2.06·26-s − 27-s + 5.18·28-s + 2.30·29-s − 1.09·31-s + 3.52·32-s − 4.67·33-s + 5.31·34-s + ⋯ |
| L(s) = 1 | − 1.67·2-s − 0.577·3-s + 1.79·4-s + 0.964·6-s + 0.547·7-s − 1.32·8-s + 0.333·9-s + 1.40·11-s − 1.03·12-s + 0.242·13-s − 0.914·14-s + 0.419·16-s − 0.545·17-s − 0.556·18-s − 1.29·19-s − 0.315·21-s − 2.35·22-s − 1.12·23-s + 0.764·24-s − 0.404·26-s − 0.192·27-s + 0.980·28-s + 0.427·29-s − 0.196·31-s + 0.622·32-s − 0.813·33-s + 0.911·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 107 | \( 1 + T \) |
| good | 2 | \( 1 + 2.36T + 2T^{2} \) |
| 7 | \( 1 - 1.44T + 7T^{2} \) |
| 11 | \( 1 - 4.67T + 11T^{2} \) |
| 13 | \( 1 - 0.873T + 13T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 + 5.64T + 19T^{2} \) |
| 23 | \( 1 + 5.37T + 23T^{2} \) |
| 29 | \( 1 - 2.30T + 29T^{2} \) |
| 31 | \( 1 + 1.09T + 31T^{2} \) |
| 37 | \( 1 - 6.05T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 4.88T + 43T^{2} \) |
| 47 | \( 1 + 0.209T + 47T^{2} \) |
| 53 | \( 1 - 8.83T + 53T^{2} \) |
| 59 | \( 1 - 2.79T + 59T^{2} \) |
| 61 | \( 1 + 8.08T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 + 0.698T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 - 2.10T + 89T^{2} \) |
| 97 | \( 1 - 3.68T + 97T^{2} \) |
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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
−7.57319736404637465638963113745, −6.97988433163655795539440245192, −6.21160147194752417018587777234, −5.90348224060174470874740895506, −4.34822053983163592317694431297, −4.22632921739240239837470517219, −2.65147924892041909739266488004, −1.77015573411027350682075702364, −1.12153054381210962136775768473, 0,
1.12153054381210962136775768473, 1.77015573411027350682075702364, 2.65147924892041909739266488004, 4.22632921739240239837470517219, 4.34822053983163592317694431297, 5.90348224060174470874740895506, 6.21160147194752417018587777234, 6.97988433163655795539440245192, 7.57319736404637465638963113745
