| L(s) = 1 | + 0.551·2-s − 1.69·4-s − 0.574·7-s − 2.03·8-s + 3.83·11-s − 2·13-s − 0.316·14-s + 2.27·16-s − 5.17·17-s + 19-s + 2.11·22-s + 1.33·23-s − 1.10·26-s + 0.973·28-s + 0.934·29-s + 4.96·31-s + 5.32·32-s − 2.85·34-s + 3.39·37-s + 0.551·38-s + 3.37·41-s − 4.85·43-s − 6.51·44-s + 0.736·46-s − 9.88·47-s − 6.67·49-s + 3.39·52-s + ⋯ |
| L(s) = 1 | + 0.389·2-s − 0.848·4-s − 0.217·7-s − 0.720·8-s + 1.15·11-s − 0.554·13-s − 0.0845·14-s + 0.567·16-s − 1.25·17-s + 0.229·19-s + 0.450·22-s + 0.278·23-s − 0.216·26-s + 0.184·28-s + 0.173·29-s + 0.892·31-s + 0.941·32-s − 0.489·34-s + 0.557·37-s + 0.0893·38-s + 0.526·41-s − 0.739·43-s − 0.981·44-s + 0.108·46-s − 1.44·47-s − 0.952·49-s + 0.470·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.551T + 2T^{2} \) |
| 7 | \( 1 + 0.574T + 7T^{2} \) |
| 11 | \( 1 - 3.83T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 5.17T + 17T^{2} \) |
| 23 | \( 1 - 1.33T + 23T^{2} \) |
| 29 | \( 1 - 0.934T + 29T^{2} \) |
| 31 | \( 1 - 4.96T + 31T^{2} \) |
| 37 | \( 1 - 3.39T + 37T^{2} \) |
| 41 | \( 1 - 3.37T + 41T^{2} \) |
| 43 | \( 1 + 4.85T + 43T^{2} \) |
| 47 | \( 1 + 9.88T + 47T^{2} \) |
| 53 | \( 1 - 3.13T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 - 8.08T + 61T^{2} \) |
| 67 | \( 1 - 0.115T + 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 + 5.85T + 73T^{2} \) |
| 79 | \( 1 + 8.35T + 79T^{2} \) |
| 83 | \( 1 - 6.51T + 83T^{2} \) |
| 89 | \( 1 + 3.13T + 89T^{2} \) |
| 97 | \( 1 + 9.68T + 97T^{2} \) |
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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
−8.242017424085399140868582214763, −7.13581995265060785719617465553, −6.49459763894374992902023228875, −5.81510818305467165415796894400, −4.76922649616668110404764665987, −4.40175321855695905493862125145, −3.50933684604557562860904312392, −2.65182494579829915495011004080, −1.32466187004907691538257100221, 0,
1.32466187004907691538257100221, 2.65182494579829915495011004080, 3.50933684604557562860904312392, 4.40175321855695905493862125145, 4.76922649616668110404764665987, 5.81510818305467165415796894400, 6.49459763894374992902023228875, 7.13581995265060785719617465553, 8.242017424085399140868582214763
