| L(s) = 1 | − 2-s − 3-s + 4-s − 2.25·5-s + 6-s + 1.35·7-s − 8-s + 9-s + 2.25·10-s + 1.14·11-s − 12-s + 0.110·13-s − 1.35·14-s + 2.25·15-s + 16-s + 7.57·17-s − 18-s − 7.57·19-s − 2.25·20-s − 1.35·21-s − 1.14·22-s − 23-s + 24-s + 0.0989·25-s − 0.110·26-s − 27-s + 1.35·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.00·5-s + 0.408·6-s + 0.512·7-s − 0.353·8-s + 0.333·9-s + 0.714·10-s + 0.346·11-s − 0.288·12-s + 0.0306·13-s − 0.362·14-s + 0.583·15-s + 0.250·16-s + 1.83·17-s − 0.235·18-s − 1.73·19-s − 0.504·20-s − 0.295·21-s − 0.244·22-s − 0.208·23-s + 0.204·24-s + 0.0197·25-s − 0.0216·26-s − 0.192·27-s + 0.256·28-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 | \( 1 + 2.25T + 5T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 - 0.110T + 13T^{2} \) |
| 17 | \( 1 - 7.57T + 17T^{2} \) |
| 19 | \( 1 + 7.57T + 19T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 6.08T + 41T^{2} \) |
| 43 | \( 1 + 7.97T + 43T^{2} \) |
| 47 | \( 1 - 4.67T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 + 5.32T + 67T^{2} \) |
| 71 | \( 1 + 1.59T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 0.564T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 0.540T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−8.171874366711058341682786789228, −7.42912316872687614618812014257, −6.83273811015277338609180970804, −5.88859825357200398995746609355, −5.21957185567357356586545133267, −4.09950670220688562053097865653, −3.60961268161993976456194889568, −2.21989771787079044870588828463, −1.15985899128314676835907903095, 0,
1.15985899128314676835907903095, 2.21989771787079044870588828463, 3.60961268161993976456194889568, 4.09950670220688562053097865653, 5.21957185567357356586545133267, 5.88859825357200398995746609355, 6.83273811015277338609180970804, 7.42912316872687614618812014257, 8.171874366711058341682786789228
