| L(s) = 1 | − 8.95·3-s + 3.69·5-s − 1.47·7-s + 53.2·9-s + 59.1·11-s + 12.0·13-s − 33.0·15-s − 42.1·19-s + 13.1·21-s + 99.0·23-s − 111.·25-s − 235.·27-s − 254.·29-s − 168.·31-s − 530.·33-s − 5.43·35-s + 76.8·37-s − 108.·39-s + 47.1·41-s − 227.·43-s + 196.·45-s + 279.·47-s − 340.·49-s − 211.·53-s + 218.·55-s + 377.·57-s + 105.·59-s + ⋯ |
| L(s) = 1 | − 1.72·3-s + 0.330·5-s − 0.0795·7-s + 1.97·9-s + 1.62·11-s + 0.257·13-s − 0.569·15-s − 0.509·19-s + 0.137·21-s + 0.897·23-s − 0.890·25-s − 1.67·27-s − 1.63·29-s − 0.977·31-s − 2.79·33-s − 0.0262·35-s + 0.341·37-s − 0.444·39-s + 0.179·41-s − 0.807·43-s + 0.651·45-s + 0.868·47-s − 0.993·49-s − 0.547·53-s + 0.535·55-s + 0.877·57-s + 0.232·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 8.95T + 27T^{2} \) |
| 5 | \( 1 - 3.69T + 125T^{2} \) |
| 7 | \( 1 + 1.47T + 343T^{2} \) |
| 11 | \( 1 - 59.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 12.0T + 2.19e3T^{2} \) |
| 19 | \( 1 + 42.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 99.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 254.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 76.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 47.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 227.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 279.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 211.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 105.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 826.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 638.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.08e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 142.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 218.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 172.T + 9.12e5T^{2} \) |
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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
−8.253156719412276907937508682621, −6.98015661564458053150272094411, −6.73336064162697709674155022878, −5.80780981241668806798639781506, −5.38764460629424415101625418471, −4.29019285084622511827611475869, −3.66898869547902649401732349450, −1.90618755862558435376369626908, −1.09062677190107354917318450830, 0,
1.09062677190107354917318450830, 1.90618755862558435376369626908, 3.66898869547902649401732349450, 4.29019285084622511827611475869, 5.38764460629424415101625418471, 5.80780981241668806798639781506, 6.73336064162697709674155022878, 6.98015661564458053150272094411, 8.253156719412276907937508682621
