| L(s) = 1 | + 0.670·2-s − 7.55·4-s − 9.86·5-s − 10.4·8-s − 6.61·10-s − 13.5·11-s + 36.6·13-s + 53.4·16-s + 100.·17-s − 36.2·19-s + 74.5·20-s − 9.10·22-s − 125.·23-s − 27.6·25-s + 24.5·26-s − 153.·29-s + 224.·31-s + 119.·32-s + 67.3·34-s + 172.·37-s − 24.2·38-s + 102.·40-s + 349.·41-s + 302.·43-s + 102.·44-s − 84.3·46-s − 5.42·47-s + ⋯ |
| L(s) = 1 | + 0.237·2-s − 0.943·4-s − 0.882·5-s − 0.460·8-s − 0.209·10-s − 0.372·11-s + 0.781·13-s + 0.834·16-s + 1.43·17-s − 0.437·19-s + 0.833·20-s − 0.0882·22-s − 1.14·23-s − 0.220·25-s + 0.185·26-s − 0.983·29-s + 1.30·31-s + 0.658·32-s + 0.339·34-s + 0.766·37-s − 0.103·38-s + 0.406·40-s + 1.33·41-s + 1.07·43-s + 0.351·44-s − 0.270·46-s − 0.0168·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.670T + 8T^{2} \) |
| 5 | \( 1 + 9.86T + 125T^{2} \) |
| 11 | \( 1 + 13.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 100.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 36.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 153.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 224.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 172.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 349.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 302.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 5.42T + 1.03e5T^{2} \) |
| 53 | \( 1 - 580.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 88.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 656.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 331.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 526.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 16.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 115.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 197.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.17e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.47e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.754035506366080216265068919111, −7.978835468840183714817394898841, −7.57195087815262400420069797832, −6.07542028216604139791182587971, −5.54848807996728466353301088434, −4.27150082612588948209237654646, −3.91527028260460197900560272540, −2.82311652552381739002718457228, −1.10572901177050253837403065059, 0,
1.10572901177050253837403065059, 2.82311652552381739002718457228, 3.91527028260460197900560272540, 4.27150082612588948209237654646, 5.54848807996728466353301088434, 6.07542028216604139791182587971, 7.57195087815262400420069797832, 7.978835468840183714817394898841, 8.754035506366080216265068919111
