| L(s) = 1 | + 1.89·2-s + 0.111·3-s + 1.58·4-s + 1.90·5-s + 0.210·6-s − 0.977·7-s − 0.794·8-s − 2.98·9-s + 3.60·10-s + 11-s + 0.175·12-s + 3.22·13-s − 1.84·14-s + 0.212·15-s − 4.66·16-s − 0.0406·17-s − 5.65·18-s − 5.58·19-s + 3.01·20-s − 0.108·21-s + 1.89·22-s + 1.35·23-s − 0.0883·24-s − 1.36·25-s + 6.09·26-s − 0.666·27-s − 1.54·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.0642·3-s + 0.790·4-s + 0.852·5-s + 0.0859·6-s − 0.369·7-s − 0.280·8-s − 0.995·9-s + 1.14·10-s + 0.301·11-s + 0.0507·12-s + 0.893·13-s − 0.494·14-s + 0.0547·15-s − 1.16·16-s − 0.00985·17-s − 1.33·18-s − 1.28·19-s + 0.673·20-s − 0.0237·21-s + 0.403·22-s + 0.283·23-s − 0.0180·24-s − 0.272·25-s + 1.19·26-s − 0.128·27-s − 0.291·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 1.89T + 2T^{2} \) |
| 3 | \( 1 - 0.111T + 3T^{2} \) |
| 5 | \( 1 - 1.90T + 5T^{2} \) |
| 7 | \( 1 + 0.977T + 7T^{2} \) |
| 13 | \( 1 - 3.22T + 13T^{2} \) |
| 17 | \( 1 + 0.0406T + 17T^{2} \) |
| 19 | \( 1 + 5.58T + 19T^{2} \) |
| 23 | \( 1 - 1.35T + 23T^{2} \) |
| 31 | \( 1 + 0.266T + 31T^{2} \) |
| 37 | \( 1 + 1.39T + 37T^{2} \) |
| 41 | \( 1 - 2.46T + 41T^{2} \) |
| 43 | \( 1 - 4.36T + 43T^{2} \) |
| 47 | \( 1 - 3.06T + 47T^{2} \) |
| 53 | \( 1 - 0.521T + 53T^{2} \) |
| 59 | \( 1 + 8.36T + 59T^{2} \) |
| 61 | \( 1 - 4.94T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 6.28T + 71T^{2} \) |
| 73 | \( 1 + 8.73T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 - 9.00T + 83T^{2} \) |
| 89 | \( 1 + 8.83T + 89T^{2} \) |
| 97 | \( 1 - 2.65T + 97T^{2} \) |
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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
−7.00285560980507782326564633966, −6.27215690553564329276071191635, −5.95503567743707208014316186298, −5.46641320902968966428662808941, −4.51163372774824573762817108193, −3.93055793226906627881615674525, −3.09388378474759386419775619519, −2.53132313107242513295743726630, −1.57966224783307389769829445024, 0,
1.57966224783307389769829445024, 2.53132313107242513295743726630, 3.09388378474759386419775619519, 3.93055793226906627881615674525, 4.51163372774824573762817108193, 5.46641320902968966428662808941, 5.95503567743707208014316186298, 6.27215690553564329276071191635, 7.00285560980507782326564633966
