L(s) = 1 | + 2·2-s − 10·3-s + 4·4-s − 20·6-s + 8·8-s + 73·9-s + 53·11-s − 40·12-s − 25·13-s + 16·16-s − 14·17-s + 146·18-s − 95·19-s + 106·22-s − 23-s − 80·24-s − 50·26-s − 460·27-s − 206·29-s + 108·31-s + 32·32-s − 530·33-s − 28·34-s + 292·36-s + 57·37-s − 190·38-s + 250·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.92·3-s + 1/2·4-s − 1.36·6-s + 0.353·8-s + 2.70·9-s + 1.45·11-s − 0.962·12-s − 0.533·13-s + 1/4·16-s − 0.199·17-s + 1.91·18-s − 1.14·19-s + 1.02·22-s − 0.00906·23-s − 0.680·24-s − 0.377·26-s − 3.27·27-s − 1.31·29-s + 0.625·31-s + 0.176·32-s − 2.79·33-s − 0.141·34-s + 1.35·36-s + 0.253·37-s − 0.811·38-s + 1.02·39-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
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
good | 3 | \( 1 + 10 T + p^{3} T^{2} \) |
| 11 | \( 1 - 53 T + p^{3} T^{2} \) |
| 13 | \( 1 + 25 T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 5 p T + p^{3} T^{2} \) |
| 23 | \( 1 + T + p^{3} T^{2} \) |
| 29 | \( 1 + 206 T + p^{3} T^{2} \) |
| 31 | \( 1 - 108 T + p^{3} T^{2} \) |
| 37 | \( 1 - 57 T + p^{3} T^{2} \) |
| 41 | \( 1 - 243 T + p^{3} T^{2} \) |
| 43 | \( 1 + 434 T + p^{3} T^{2} \) |
| 47 | \( 1 - 231 T + p^{3} T^{2} \) |
| 53 | \( 1 + 263 T + p^{3} T^{2} \) |
| 59 | \( 1 - 24 T + p^{3} T^{2} \) |
| 61 | \( 1 - 116 T + p^{3} T^{2} \) |
| 67 | \( 1 - 204 T + p^{3} T^{2} \) |
| 71 | \( 1 - 484 T + p^{3} T^{2} \) |
| 73 | \( 1 - 692 T + p^{3} T^{2} \) |
| 79 | \( 1 - 466 T + p^{3} T^{2} \) |
| 83 | \( 1 + 228 T + p^{3} T^{2} \) |
| 89 | \( 1 + 362 T + p^{3} T^{2} \) |
| 97 | \( 1 + 854 T + p^{3} T^{2} \) |
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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
−7.907367230250473392318594360826, −6.87607965801577406657530133667, −6.57003229162526716316249900321, −5.86460102007572322975612813057, −5.09920775524285571892109242985, −4.34383220161283780726321582312, −3.78462121035277861232605629933, −2.08425202719541802662812755553, −1.13143278168042238417467294541, 0,
1.13143278168042238417467294541, 2.08425202719541802662812755553, 3.78462121035277861232605629933, 4.34383220161283780726321582312, 5.09920775524285571892109242985, 5.86460102007572322975612813057, 6.57003229162526716316249900321, 6.87607965801577406657530133667, 7.907367230250473392318594360826