L(s) = 1 | − 2·2-s + 7·3-s + 4·4-s − 14·6-s − 7·7-s − 8·8-s + 22·9-s − 37·11-s + 28·12-s − 51·13-s + 14·14-s + 16·16-s − 41·17-s − 44·18-s − 108·19-s − 49·21-s + 74·22-s + 70·23-s − 56·24-s + 102·26-s − 35·27-s − 28·28-s − 249·29-s − 134·31-s − 32·32-s − 259·33-s + 82·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.34·3-s + 1/2·4-s − 0.952·6-s − 0.377·7-s − 0.353·8-s + 0.814·9-s − 1.01·11-s + 0.673·12-s − 1.08·13-s + 0.267·14-s + 1/4·16-s − 0.584·17-s − 0.576·18-s − 1.30·19-s − 0.509·21-s + 0.717·22-s + 0.634·23-s − 0.476·24-s + 0.769·26-s − 0.249·27-s − 0.188·28-s − 1.59·29-s − 0.776·31-s − 0.176·32-s − 1.36·33-s + 0.413·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 37 T + p^{3} T^{2} \) |
| 13 | \( 1 + 51 T + p^{3} T^{2} \) |
| 17 | \( 1 + 41 T + p^{3} T^{2} \) |
| 19 | \( 1 + 108 T + p^{3} T^{2} \) |
| 23 | \( 1 - 70 T + p^{3} T^{2} \) |
| 29 | \( 1 + 249 T + p^{3} T^{2} \) |
| 31 | \( 1 + 134 T + p^{3} T^{2} \) |
| 37 | \( 1 - 334 T + p^{3} T^{2} \) |
| 41 | \( 1 - 206 T + p^{3} T^{2} \) |
| 43 | \( 1 - 376 T + p^{3} T^{2} \) |
| 47 | \( 1 - 287 T + p^{3} T^{2} \) |
| 53 | \( 1 - 6 T + p^{3} T^{2} \) |
| 59 | \( 1 + 2 T + p^{3} T^{2} \) |
| 61 | \( 1 + 940 T + p^{3} T^{2} \) |
| 67 | \( 1 + 106 T + p^{3} T^{2} \) |
| 71 | \( 1 - 456 T + p^{3} T^{2} \) |
| 73 | \( 1 + 650 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1239 T + p^{3} T^{2} \) |
| 83 | \( 1 + 428 T + p^{3} T^{2} \) |
| 89 | \( 1 + 220 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1055 T + p^{3} T^{2} \) |
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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
−10.39214305996520387906339888869, −9.353799463143717572254419749900, −8.917233041536028753055887601783, −7.77195361751981983908474916846, −7.30423702925814586430420233260, −5.84980864057896464502833975805, −4.27224722524168557518000785924, −2.84053479250650156946967116642, −2.15604093861471443380110754259, 0,
2.15604093861471443380110754259, 2.84053479250650156946967116642, 4.27224722524168557518000785924, 5.84980864057896464502833975805, 7.30423702925814586430420233260, 7.77195361751981983908474916846, 8.917233041536028753055887601783, 9.353799463143717572254419749900, 10.39214305996520387906339888869