L(s) = 1 | + 2-s + 1.43·3-s + 4-s − 0.301·5-s + 1.43·6-s − 7-s + 8-s − 0.952·9-s − 0.301·10-s + 1.43·12-s − 1.39·13-s − 14-s − 0.430·15-s + 16-s − 3.90·17-s − 0.952·18-s + 19-s − 0.301·20-s − 1.43·21-s − 3.57·23-s + 1.43·24-s − 4.90·25-s − 1.39·26-s − 5.65·27-s − 28-s − 0.635·29-s − 0.430·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.826·3-s + 0.5·4-s − 0.134·5-s + 0.584·6-s − 0.377·7-s + 0.353·8-s − 0.317·9-s − 0.0952·10-s + 0.413·12-s − 0.387·13-s − 0.267·14-s − 0.111·15-s + 0.250·16-s − 0.948·17-s − 0.224·18-s + 0.229·19-s − 0.0673·20-s − 0.312·21-s − 0.745·23-s + 0.292·24-s − 0.981·25-s − 0.274·26-s − 1.08·27-s − 0.188·28-s − 0.118·29-s − 0.0786·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 1.43T + 3T^{2} \) |
| 5 | \( 1 + 0.301T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 13 | \( 1 + 1.39T + 13T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 23 | \( 1 + 3.57T + 23T^{2} \) |
| 29 | \( 1 + 0.635T + 29T^{2} \) |
| 31 | \( 1 + 8.90T + 31T^{2} \) |
| 37 | \( 1 + 0.187T + 37T^{2} \) |
| 41 | \( 1 - 8.02T + 41T^{2} \) |
| 43 | \( 1 + 0.641T + 43T^{2} \) |
| 47 | \( 1 + 7.55T + 47T^{2} \) |
| 53 | \( 1 + 0.919T + 53T^{2} \) |
| 59 | \( 1 - 0.502T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 - 8.14T + 67T^{2} \) |
| 71 | \( 1 - 7.20T + 71T^{2} \) |
| 73 | \( 1 - 4.86T + 73T^{2} \) |
| 79 | \( 1 - 5.73T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 - 3.76T + 89T^{2} \) |
| 97 | \( 1 + 1.58T + 97T^{2} \) |
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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
−7.888762368225341668302141772883, −7.30292257462707207402335796907, −6.42260747132261012961720211580, −5.76004060769537348799308084524, −4.93291305290375171475945013196, −3.96186087940494494202280725326, −3.46146504019277199382193284731, −2.51612667684021979583241287451, −1.87755396370102537694822600024, 0,
1.87755396370102537694822600024, 2.51612667684021979583241287451, 3.46146504019277199382193284731, 3.96186087940494494202280725326, 4.93291305290375171475945013196, 5.76004060769537348799308084524, 6.42260747132261012961720211580, 7.30292257462707207402335796907, 7.888762368225341668302141772883