| L(s) = 1 | − 2·2-s − 7·3-s + 4·4-s + 14·6-s − 8·8-s + 22·9-s − 37·11-s − 28·12-s + 51·13-s + 16·16-s + 41·17-s − 44·18-s + 108·19-s + 74·22-s + 70·23-s + 56·24-s − 102·26-s + 35·27-s − 249·29-s + 134·31-s − 32·32-s + 259·33-s − 82·34-s + 88·36-s + 334·37-s − 216·38-s − 357·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·3-s + 1/2·4-s + 0.952·6-s − 0.353·8-s + 0.814·9-s − 1.01·11-s − 0.673·12-s + 1.08·13-s + 1/4·16-s + 0.584·17-s − 0.576·18-s + 1.30·19-s + 0.717·22-s + 0.634·23-s + 0.476·24-s − 0.769·26-s + 0.249·27-s − 1.59·29-s + 0.776·31-s − 0.176·32-s + 1.36·33-s − 0.413·34-s + 0.407·36-s + 1.48·37-s − 0.922·38-s − 1.46·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9003516372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9003516372\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 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
−8.527259785355391865168641107197, −7.78059352138516444045297451606, −7.09714703935566622624042169251, −6.17827302025823209895203088113, −5.58797735565882411243560914945, −4.99505154327013472725489095979, −3.70246339185228506312641729372, −2.67398808471436213284214440920, −1.27665075767756063101538773976, −0.57345904786561528599893467382,
0.57345904786561528599893467382, 1.27665075767756063101538773976, 2.67398808471436213284214440920, 3.70246339185228506312641729372, 4.99505154327013472725489095979, 5.58797735565882411243560914945, 6.17827302025823209895203088113, 7.09714703935566622624042169251, 7.78059352138516444045297451606, 8.527259785355391865168641107197
