L(s) = 1 | − 2-s + 3.10·3-s + 4-s + 5-s − 3.10·6-s − 3.10·7-s − 8-s + 6.61·9-s − 10-s − 11-s + 3.10·12-s − 4·13-s + 3.10·14-s + 3.10·15-s + 16-s − 4.58·17-s − 6.61·18-s − 3.61·19-s + 20-s − 9.61·21-s + 22-s − 3.10·24-s + 25-s + 4·26-s + 11.2·27-s − 3.10·28-s + 2.48·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.79·3-s + 0.5·4-s + 0.447·5-s − 1.26·6-s − 1.17·7-s − 0.353·8-s + 2.20·9-s − 0.316·10-s − 0.301·11-s + 0.895·12-s − 1.10·13-s + 0.828·14-s + 0.800·15-s + 0.250·16-s − 1.11·17-s − 1.55·18-s − 0.828·19-s + 0.223·20-s − 2.09·21-s + 0.213·22-s − 0.632·24-s + 0.200·25-s + 0.784·26-s + 2.15·27-s − 0.585·28-s + 0.461·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
good | 3 | \( 1 - 3.10T + 3T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2.48T + 29T^{2} \) |
| 31 | \( 1 + 5.71T + 31T^{2} \) |
| 37 | \( 1 - 2.68T + 37T^{2} \) |
| 41 | \( 1 + 4.20T + 41T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 + 6.63T + 53T^{2} \) |
| 59 | \( 1 + 8.61T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 0.899T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 1.43T + 79T^{2} \) |
| 83 | \( 1 + 5.43T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 9.02T + 97T^{2} \) |
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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
−8.132489018813222694961573760291, −7.27280298345585766024471790017, −6.86440263462481472875941559094, −6.05284211912560284508392623198, −4.77689083459532606200677268968, −3.89309999731543706146947659273, −2.94739732753632036215248803871, −2.48941168702190012581220229182, −1.72813143680326724805799009293, 0,
1.72813143680326724805799009293, 2.48941168702190012581220229182, 2.94739732753632036215248803871, 3.89309999731543706146947659273, 4.77689083459532606200677268968, 6.05284211912560284508392623198, 6.86440263462481472875941559094, 7.27280298345585766024471790017, 8.132489018813222694961573760291