| L(s) = 1 | − 2-s − 2.44·3-s + 4-s + 2.44·6-s − 2.44·7-s − 8-s + 2.99·9-s − 11-s − 2.44·12-s + 3.89·13-s + 2.44·14-s + 16-s − 4.44·17-s − 2.99·18-s + 5.99·21-s + 22-s + 0.898·23-s + 2.44·24-s − 5·25-s − 3.89·26-s − 2.44·28-s − 1.89·29-s + 2·31-s − 32-s + 2.44·33-s + 4.44·34-s + 2.99·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.41·3-s + 0.5·4-s + 0.999·6-s − 0.925·7-s − 0.353·8-s + 0.999·9-s − 0.301·11-s − 0.707·12-s + 1.08·13-s + 0.654·14-s + 0.250·16-s − 1.07·17-s − 0.707·18-s + 1.30·21-s + 0.213·22-s + 0.187·23-s + 0.499·24-s − 25-s − 0.764·26-s − 0.462·28-s − 0.352·29-s + 0.359·31-s − 0.176·32-s + 0.426·33-s + 0.763·34-s + 0.499·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 13 | \( 1 - 3.89T + 13T^{2} \) |
| 17 | \( 1 + 4.44T + 17T^{2} \) |
| 23 | \( 1 - 0.898T + 23T^{2} \) |
| 29 | \( 1 + 1.89T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 8.44T + 37T^{2} \) |
| 41 | \( 1 + 7.55T + 41T^{2} \) |
| 43 | \( 1 - 9.44T + 43T^{2} \) |
| 47 | \( 1 + 6.55T + 47T^{2} \) |
| 53 | \( 1 + 7.79T + 53T^{2} \) |
| 59 | \( 1 - 2.89T + 59T^{2} \) |
| 61 | \( 1 - 7.89T + 61T^{2} \) |
| 67 | \( 1 - 9.55T + 67T^{2} \) |
| 71 | \( 1 - 7.44T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 4.44T + 79T^{2} \) |
| 83 | \( 1 + 5.44T + 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 - 3.89T + 97T^{2} \) |
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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
−7.35037309584281506649581864380, −6.62985388894284641317021845969, −6.19181384437433359119976468970, −5.73350518428279785657722343050, −4.80341729395932522622836360021, −3.96250143451072115134435815326, −3.05078388503348811126931912402, −1.98568635131221858174912307971, −0.853207073274883459326526730479, 0,
0.853207073274883459326526730479, 1.98568635131221858174912307971, 3.05078388503348811126931912402, 3.96250143451072115134435815326, 4.80341729395932522622836360021, 5.73350518428279785657722343050, 6.19181384437433359119976468970, 6.62985388894284641317021845969, 7.35037309584281506649581864380
