| L(s) = 1 | − 2-s − 3-s + 4-s − 2.88·5-s + 6-s − 3.71·7-s − 8-s + 9-s + 2.88·10-s + 5.50·11-s − 12-s − 3.61·13-s + 3.71·14-s + 2.88·15-s + 16-s − 4.95·17-s − 18-s + 4.95·19-s − 2.88·20-s + 3.71·21-s − 5.50·22-s − 23-s + 24-s + 3.33·25-s + 3.61·26-s − 27-s − 3.71·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.29·5-s + 0.408·6-s − 1.40·7-s − 0.353·8-s + 0.333·9-s + 0.913·10-s + 1.66·11-s − 0.288·12-s − 1.00·13-s + 0.993·14-s + 0.745·15-s + 0.250·16-s − 1.20·17-s − 0.235·18-s + 1.13·19-s − 0.645·20-s + 0.810·21-s − 1.17·22-s − 0.208·23-s + 0.204·24-s + 0.667·25-s + 0.709·26-s − 0.192·27-s − 0.702·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 2.88T + 5T^{2} \) |
| 7 | \( 1 + 3.71T + 7T^{2} \) |
| 11 | \( 1 - 5.50T + 11T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 + 4.95T + 17T^{2} \) |
| 19 | \( 1 - 4.95T + 19T^{2} \) |
| 31 | \( 1 - 6.02T + 31T^{2} \) |
| 37 | \( 1 + 1.34T + 37T^{2} \) |
| 41 | \( 1 + 1.77T + 41T^{2} \) |
| 43 | \( 1 + 7.39T + 43T^{2} \) |
| 47 | \( 1 - 13.6T + 47T^{2} \) |
| 53 | \( 1 - 2.70T + 53T^{2} \) |
| 59 | \( 1 - 8.87T + 59T^{2} \) |
| 61 | \( 1 + 5.99T + 61T^{2} \) |
| 67 | \( 1 - 9.67T + 67T^{2} \) |
| 71 | \( 1 - 3.39T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 + 4.99T + 79T^{2} \) |
| 83 | \( 1 - 9.98T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 6.70T + 97T^{2} \) |
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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
−8.056733426744796758800390779368, −7.16834339684422802381238351910, −6.80036613043399725917409759551, −6.23292262448878159128716463037, −5.07698801855859205716715204747, −4.04358760341228978143757835957, −3.55835338934226451363465026818, −2.44988023498937485379709386146, −0.937170139549211323205395888021, 0,
0.937170139549211323205395888021, 2.44988023498937485379709386146, 3.55835338934226451363465026818, 4.04358760341228978143757835957, 5.07698801855859205716715204747, 6.23292262448878159128716463037, 6.80036613043399725917409759551, 7.16834339684422802381238351910, 8.056733426744796758800390779368
