L(s) = 1 | − 1.76·2-s − 2.32·3-s + 1.12·4-s − 4.06·5-s + 4.11·6-s + 1.54·8-s + 2.41·9-s + 7.18·10-s − 2.90·11-s − 2.61·12-s + 9.46·15-s − 4.98·16-s − 3.37·17-s − 4.27·18-s + 1.18·19-s − 4.56·20-s + 5.13·22-s − 0.00746·23-s − 3.60·24-s + 11.5·25-s + 1.35·27-s − 8.00·29-s − 16.7·30-s + 0.679·31-s + 5.71·32-s + 6.76·33-s + 5.96·34-s + ⋯ |
L(s) = 1 | − 1.24·2-s − 1.34·3-s + 0.561·4-s − 1.81·5-s + 1.67·6-s + 0.547·8-s + 0.806·9-s + 2.27·10-s − 0.876·11-s − 0.754·12-s + 2.44·15-s − 1.24·16-s − 0.818·17-s − 1.00·18-s + 0.272·19-s − 1.02·20-s + 1.09·22-s − 0.00155·23-s − 0.736·24-s + 2.30·25-s + 0.259·27-s − 1.48·29-s − 3.05·30-s + 0.122·31-s + 1.00·32-s + 1.17·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.76T + 2T^{2} \) |
| 3 | \( 1 + 2.32T + 3T^{2} \) |
| 5 | \( 1 + 4.06T + 5T^{2} \) |
| 11 | \( 1 + 2.90T + 11T^{2} \) |
| 17 | \( 1 + 3.37T + 17T^{2} \) |
| 19 | \( 1 - 1.18T + 19T^{2} \) |
| 23 | \( 1 + 0.00746T + 23T^{2} \) |
| 29 | \( 1 + 8.00T + 29T^{2} \) |
| 31 | \( 1 - 0.679T + 31T^{2} \) |
| 37 | \( 1 + 2.58T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 4.33T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 6.09T + 53T^{2} \) |
| 59 | \( 1 + 3.95T + 59T^{2} \) |
| 61 | \( 1 - 3.60T + 61T^{2} \) |
| 67 | \( 1 - 1.67T + 67T^{2} \) |
| 71 | \( 1 - 7.13T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 1.82T + 83T^{2} \) |
| 89 | \( 1 - 3.60T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−7.50601946171800737533294363958, −7.03889872519656687642245502193, −6.35197073365614467706023962818, −5.20589052859451508645365410956, −4.80478533270673612253630918312, −4.02364150659001441802692788724, −3.11780405876963678914633156828, −1.76317515464497307783894594523, −0.55261232962706586016733319220, 0,
0.55261232962706586016733319220, 1.76317515464497307783894594523, 3.11780405876963678914633156828, 4.02364150659001441802692788724, 4.80478533270673612253630918312, 5.20589052859451508645365410956, 6.35197073365614467706023962818, 7.03889872519656687642245502193, 7.50601946171800737533294363958