| L(s) = 1 | − 11.0·2-s + 26.0·3-s + 90.7·4-s − 288.·6-s − 153.·7-s − 651.·8-s + 435.·9-s − 121·11-s + 2.36e3·12-s + 20.2·13-s + 1.70e3·14-s + 4.30e3·16-s + 604.·17-s − 4.83e3·18-s − 858.·19-s − 4.00e3·21-s + 1.34e3·22-s + 1.57e3·23-s − 1.69e4·24-s − 223.·26-s + 5.02e3·27-s − 1.39e4·28-s − 547.·29-s − 1.06e4·31-s − 2.69e4·32-s − 3.15e3·33-s − 6.70e3·34-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 1.67·3-s + 2.83·4-s − 3.27·6-s − 1.18·7-s − 3.59·8-s + 1.79·9-s − 0.301·11-s + 4.74·12-s + 0.0331·13-s + 2.32·14-s + 4.20·16-s + 0.507·17-s − 3.51·18-s − 0.545·19-s − 1.97·21-s + 0.590·22-s + 0.619·23-s − 6.01·24-s − 0.0649·26-s + 1.32·27-s − 3.35·28-s − 0.120·29-s − 1.98·31-s − 4.64·32-s − 0.503·33-s − 0.994·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 11.0T + 32T^{2} \) |
| 3 | \( 1 - 26.0T + 243T^{2} \) |
| 7 | \( 1 + 153.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 20.2T + 3.71e5T^{2} \) |
| 17 | \( 1 - 604.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 858.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.57e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 547.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.06e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.77e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.78e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 358.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.12e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.37e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.68e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.17e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.04e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.64e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.64e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.37e4T + 8.58e9T^{2} \) |
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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
−10.10359071586888965139288922318, −9.268174850192702337738412448701, −8.951486924675714813957672716159, −7.83106655716518644554382636184, −7.27577090181133384709588911814, −6.16318354325543341749570647034, −3.44252621982081726267962433895, −2.70140870381416969540530323544, −1.57752431330013787988580189549, 0,
1.57752431330013787988580189549, 2.70140870381416969540530323544, 3.44252621982081726267962433895, 6.16318354325543341749570647034, 7.27577090181133384709588911814, 7.83106655716518644554382636184, 8.951486924675714813957672716159, 9.268174850192702337738412448701, 10.10359071586888965139288922318
