| L(s) = 1 | + 2·2-s − 5.92·3-s + 4·4-s − 11.8·6-s + 8·8-s + 8.15·9-s − 56.0·11-s − 23.7·12-s + 5.05·13-s + 16·16-s − 12.9·17-s + 16.3·18-s + 79.8·19-s − 112.·22-s + 32.0·23-s − 47.4·24-s + 10.1·26-s + 111.·27-s + 256.·29-s − 36.4·31-s + 32·32-s + 332.·33-s − 25.8·34-s + 32.6·36-s − 136.·37-s + 159.·38-s − 30·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.14·3-s + 0.5·4-s − 0.806·6-s + 0.353·8-s + 0.302·9-s − 1.53·11-s − 0.570·12-s + 0.107·13-s + 0.250·16-s − 0.184·17-s + 0.213·18-s + 0.964·19-s − 1.08·22-s + 0.290·23-s − 0.403·24-s + 0.0763·26-s + 0.796·27-s + 1.63·29-s − 0.210·31-s + 0.176·32-s + 1.75·33-s − 0.130·34-s + 0.151·36-s − 0.604·37-s + 0.681·38-s − 0.123·39-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 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 5.92T + 27T^{2} \) |
| 11 | \( 1 + 56.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 5.05T + 2.19e3T^{2} \) |
| 17 | \( 1 + 12.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 79.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 32.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 256.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 36.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 136.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 130.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 91.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 2.06T + 1.03e5T^{2} \) |
| 53 | \( 1 + 499.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 349.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 92.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 275.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 540.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 30.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.29e3T + 9.12e5T^{2} \) |
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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
−7.995311222936830130014027348768, −7.26625253511904142163807383677, −6.42954304084103259245796120587, −5.71424730590282049026544272913, −5.10232524128870730889978850231, −4.56307044773957593183740870472, −3.25025109385979407067233272187, −2.51062142415158781051942724615, −1.09687637247507047701877106814, 0,
1.09687637247507047701877106814, 2.51062142415158781051942724615, 3.25025109385979407067233272187, 4.56307044773957593183740870472, 5.10232524128870730889978850231, 5.71424730590282049026544272913, 6.42954304084103259245796120587, 7.26625253511904142163807383677, 7.995311222936830130014027348768
