| L(s) = 1 | − 2.42·2-s + 0.490·3-s + 3.85·4-s − 1.18·6-s − 7-s − 4.50·8-s − 2.75·9-s + 0.841·11-s + 1.89·12-s − 4.60·13-s + 2.42·14-s + 3.17·16-s + 1.35·17-s + 6.68·18-s + 7.24·19-s − 0.490·21-s − 2.03·22-s − 23-s − 2.20·24-s + 11.1·26-s − 2.82·27-s − 3.85·28-s − 4.95·29-s − 7.41·31-s + 1.31·32-s + 0.412·33-s − 3.28·34-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 0.283·3-s + 1.92·4-s − 0.484·6-s − 0.377·7-s − 1.59·8-s − 0.919·9-s + 0.253·11-s + 0.546·12-s − 1.27·13-s + 0.646·14-s + 0.793·16-s + 0.329·17-s + 1.57·18-s + 1.66·19-s − 0.106·21-s − 0.434·22-s − 0.208·23-s − 0.450·24-s + 2.18·26-s − 0.543·27-s − 0.729·28-s − 0.919·29-s − 1.33·31-s + 0.232·32-s + 0.0718·33-s − 0.563·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.42T + 2T^{2} \) |
| 3 | \( 1 - 0.490T + 3T^{2} \) |
| 11 | \( 1 - 0.841T + 11T^{2} \) |
| 13 | \( 1 + 4.60T + 13T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 - 7.24T + 19T^{2} \) |
| 29 | \( 1 + 4.95T + 29T^{2} \) |
| 31 | \( 1 + 7.41T + 31T^{2} \) |
| 37 | \( 1 - 8.07T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 8.95T + 53T^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 + 8.83T + 61T^{2} \) |
| 67 | \( 1 - 6.29T + 67T^{2} \) |
| 71 | \( 1 - 9.82T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 5.18T + 79T^{2} \) |
| 83 | \( 1 - 1.89T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 8.69T + 97T^{2} \) |
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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
−7.983648891466147093998585818261, −7.58564102723588565660071314357, −7.11024816004622892211968887032, −5.97486316849854563093456547156, −5.43622294577363999742784031300, −4.06362175474252110151471258423, −2.89691232572463923814422686856, −2.38348743198399641352058403353, −1.12192236440378449676348192418, 0,
1.12192236440378449676348192418, 2.38348743198399641352058403353, 2.89691232572463923814422686856, 4.06362175474252110151471258423, 5.43622294577363999742784031300, 5.97486316849854563093456547156, 7.11024816004622892211968887032, 7.58564102723588565660071314357, 7.983648891466147093998585818261
