| L(s) = 1 | − 1.41·2-s − 1.41·3-s + 2.82·5-s + 2.00·6-s + 2.82·8-s − 0.999·9-s − 4.00·10-s − 0.171·11-s − 6.24·13-s − 4.00·15-s − 4.00·16-s + 3·17-s + 1.41·18-s − 19-s + 0.242·22-s + 3·23-s − 4·24-s + 3.00·25-s + 8.82·26-s + 5.65·27-s − 3.17·29-s + 5.65·30-s + 2.24·31-s + 0.242·33-s − 4.24·34-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 0.816·3-s + 1.26·5-s + 0.816·6-s + 0.999·8-s − 0.333·9-s − 1.26·10-s − 0.0517·11-s − 1.73·13-s − 1.03·15-s − 1.00·16-s + 0.727·17-s + 0.333·18-s − 0.229·19-s + 0.0517·22-s + 0.625·23-s − 0.816·24-s + 0.600·25-s + 1.73·26-s + 1.08·27-s − 0.588·29-s + 1.03·30-s + 0.402·31-s + 0.0422·33-s − 0.727·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 + 0.171T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 - 2.24T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 6.17T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 3T + 73T^{2} \) |
| 79 | \( 1 - 3.75T + 79T^{2} \) |
| 83 | \( 1 - 5.48T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 6.24T + 97T^{2} \) |
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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
−9.576229216755299219097243294837, −9.174929210387777506839028358619, −8.020440537816109065096975884640, −7.19521945446001374980781331492, −6.18975487041614246394805611945, −5.31088588791832216342589815201, −4.68938215284922988322314899467, −2.77550211802453478616425428304, −1.53966979131045671703248730355, 0,
1.53966979131045671703248730355, 2.77550211802453478616425428304, 4.68938215284922988322314899467, 5.31088588791832216342589815201, 6.18975487041614246394805611945, 7.19521945446001374980781331492, 8.020440537816109065096975884640, 9.174929210387777506839028358619, 9.576229216755299219097243294837
