L(s) = 1 | + 2-s − 0.178·3-s + 4-s − 1.41·5-s − 0.178·6-s − 1.81·7-s + 8-s − 2.96·9-s − 1.41·10-s + 2.27·11-s − 0.178·12-s + 4.66·13-s − 1.81·14-s + 0.251·15-s + 16-s − 3.31·17-s − 2.96·18-s + 0.914·19-s − 1.41·20-s + 0.323·21-s + 2.27·22-s + 23-s − 0.178·24-s − 3.00·25-s + 4.66·26-s + 1.06·27-s − 1.81·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.102·3-s + 0.5·4-s − 0.631·5-s − 0.0727·6-s − 0.686·7-s + 0.353·8-s − 0.989·9-s − 0.446·10-s + 0.685·11-s − 0.0514·12-s + 1.29·13-s − 0.485·14-s + 0.0649·15-s + 0.250·16-s − 0.804·17-s − 0.699·18-s + 0.209·19-s − 0.315·20-s + 0.0705·21-s + 0.484·22-s + 0.208·23-s − 0.0363·24-s − 0.600·25-s + 0.914·26-s + 0.204·27-s − 0.343·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 + 0.178T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 1.81T + 7T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 13 | \( 1 - 4.66T + 13T^{2} \) |
| 17 | \( 1 + 3.31T + 17T^{2} \) |
| 19 | \( 1 - 0.914T + 19T^{2} \) |
| 29 | \( 1 + 1.00T + 29T^{2} \) |
| 31 | \( 1 - 3.63T + 31T^{2} \) |
| 37 | \( 1 - 5.78T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 2.18T + 43T^{2} \) |
| 47 | \( 1 - 3.07T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 + 7.70T + 59T^{2} \) |
| 61 | \( 1 - 8.68T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 - 2.58T + 71T^{2} \) |
| 73 | \( 1 + 3.33T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 8.79T + 97T^{2} \) |
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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
−7.65964654922087753117093616132, −6.74196007392613620104679597627, −6.27624852912374717470847107226, −5.70338125743672105311606469713, −4.73816076186414625954548310772, −3.91945598185355368141863100706, −3.40834721294422073971780641129, −2.62315562177100957142623282988, −1.36950521045049169998370783175, 0,
1.36950521045049169998370783175, 2.62315562177100957142623282988, 3.40834721294422073971780641129, 3.91945598185355368141863100706, 4.73816076186414625954548310772, 5.70338125743672105311606469713, 6.27624852912374717470847107226, 6.74196007392613620104679597627, 7.65964654922087753117093616132