L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 0.979·11-s + 12-s + 0.435·13-s + 16-s − 2.79·17-s − 18-s − 7.34·19-s − 0.979·22-s + 3.34·23-s − 24-s − 0.435·26-s + 27-s + 3.74·29-s − 4.97·31-s − 32-s + 0.979·33-s + 2.79·34-s + 36-s − 4.63·37-s + 7.34·38-s + 0.435·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.295·11-s + 0.288·12-s + 0.120·13-s + 0.250·16-s − 0.678·17-s − 0.235·18-s − 1.68·19-s − 0.208·22-s + 0.697·23-s − 0.204·24-s − 0.0853·26-s + 0.192·27-s + 0.696·29-s − 0.894·31-s − 0.176·32-s + 0.170·33-s + 0.479·34-s + 0.166·36-s − 0.762·37-s + 1.19·38-s + 0.0696·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 0.979T + 11T^{2} \) |
| 13 | \( 1 - 0.435T + 13T^{2} \) |
| 17 | \( 1 + 2.79T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 23 | \( 1 - 3.34T + 23T^{2} \) |
| 29 | \( 1 - 3.74T + 29T^{2} \) |
| 31 | \( 1 + 4.97T + 31T^{2} \) |
| 37 | \( 1 + 4.63T + 37T^{2} \) |
| 41 | \( 1 + 4.94T + 41T^{2} \) |
| 43 | \( 1 - 9.97T + 43T^{2} \) |
| 47 | \( 1 - 4.40T + 47T^{2} \) |
| 53 | \( 1 + 0.657T + 53T^{2} \) |
| 59 | \( 1 + 8.27T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 2.36T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 2.88T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.62998000927034584391976294956, −6.94803181504043424266370279287, −6.43570324739290244122443082881, −5.58756455984735381105536445921, −4.53457380945786345308106466451, −3.92097614507486553720108025008, −2.92422168176050962834127342824, −2.19370679485331500665076474359, −1.34262038901469816155250849376, 0,
1.34262038901469816155250849376, 2.19370679485331500665076474359, 2.92422168176050962834127342824, 3.92097614507486553720108025008, 4.53457380945786345308106466451, 5.58756455984735381105536445921, 6.43570324739290244122443082881, 6.94803181504043424266370279287, 7.62998000927034584391976294956