| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s − 0.763·11-s − 12-s + 1.85·13-s − 2·14-s + 16-s + 1.14·17-s − 18-s − 7.23·19-s − 2·21-s + 0.763·22-s + 6·23-s + 24-s − 1.85·26-s − 27-s + 2·28-s − 3.61·29-s − 9.70·31-s − 32-s + 0.763·33-s − 1.14·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 0.333·9-s − 0.230·11-s − 0.288·12-s + 0.514·13-s − 0.534·14-s + 0.250·16-s + 0.277·17-s − 0.235·18-s − 1.66·19-s − 0.436·21-s + 0.162·22-s + 1.25·23-s + 0.204·24-s − 0.363·26-s − 0.192·27-s + 0.377·28-s − 0.671·29-s − 1.74·31-s − 0.176·32-s + 0.132·33-s − 0.196·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 - 1.85T + 13T^{2} \) |
| 17 | \( 1 - 1.14T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 + 8.85T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 - 9.23T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 2.14T + 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 + 9.85T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 3.61T + 89T^{2} \) |
| 97 | \( 1 + 7.14T + 97T^{2} \) |
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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
−8.151982306961392511014426216207, −7.46133695483654061657086677844, −6.78070098886909779844989583588, −5.94565930321878912176040031756, −5.26045632940998153227814077550, −4.39037955941688320716652101337, −3.42111005767575422146274936702, −2.15605174263596132333072658192, −1.34906812509579783043040958521, 0,
1.34906812509579783043040958521, 2.15605174263596132333072658192, 3.42111005767575422146274936702, 4.39037955941688320716652101337, 5.26045632940998153227814077550, 5.94565930321878912176040031756, 6.78070098886909779844989583588, 7.46133695483654061657086677844, 8.151982306961392511014426216207
