| L(s) = 1 | − 2.39·2-s + 3.71·4-s − 5-s + 5.11·7-s − 4.11·8-s + 2.39·10-s − 5.43·13-s − 12.2·14-s + 2.39·16-s + 1.32·17-s − 1.67·19-s − 3.71·20-s + 2.11·23-s + 25-s + 13.0·26-s + 19.0·28-s + 0.782·29-s − 4.43·31-s + 2.50·32-s − 3.17·34-s − 5.11·35-s − 11.3·37-s + 3.99·38-s + 4.11·40-s + 11.5·41-s + 4.78·43-s − 5.04·46-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.85·4-s − 0.447·5-s + 1.93·7-s − 1.45·8-s + 0.756·10-s − 1.50·13-s − 3.26·14-s + 0.597·16-s + 0.321·17-s − 0.383·19-s − 0.831·20-s + 0.439·23-s + 0.200·25-s + 2.55·26-s + 3.59·28-s + 0.145·29-s − 0.796·31-s + 0.442·32-s − 0.544·34-s − 0.863·35-s − 1.86·37-s + 0.648·38-s + 0.649·40-s + 1.80·41-s + 0.729·43-s − 0.743·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 7 | \( 1 - 5.11T + 7T^{2} \) |
| 13 | \( 1 + 5.43T + 13T^{2} \) |
| 17 | \( 1 - 1.32T + 17T^{2} \) |
| 19 | \( 1 + 1.67T + 19T^{2} \) |
| 23 | \( 1 - 2.11T + 23T^{2} \) |
| 29 | \( 1 - 0.782T + 29T^{2} \) |
| 31 | \( 1 + 4.43T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 + 9.45T + 47T^{2} \) |
| 53 | \( 1 - 8.23T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 0.779T + 61T^{2} \) |
| 67 | \( 1 + 4.45T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 2.32T + 73T^{2} \) |
| 79 | \( 1 + 1.12T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 4.45T + 97T^{2} \) |
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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
−7.80303216304268155124113731217, −7.46214918826509213898741658090, −6.95092373477692742718354537333, −5.68111345958732323029215852734, −4.89488134119798989526097034699, −4.23995730087792830623398657733, −2.79404069400679542459000929946, −1.97201290055234085623606474675, −1.24189751622151419427497701653, 0,
1.24189751622151419427497701653, 1.97201290055234085623606474675, 2.79404069400679542459000929946, 4.23995730087792830623398657733, 4.89488134119798989526097034699, 5.68111345958732323029215852734, 6.95092373477692742718354537333, 7.46214918826509213898741658090, 7.80303216304268155124113731217
