| L(s) = 1 | + 8.14·3-s + 3.35·5-s − 27.3·7-s + 39.3·9-s + 30.4·11-s − 6.55·13-s + 27.3·15-s + 5.92·17-s − 107.·19-s − 222.·21-s + 23·23-s − 113.·25-s + 100.·27-s − 272.·29-s − 12.6·31-s + 247.·33-s − 91.6·35-s − 343.·37-s − 53.4·39-s − 80.6·41-s − 24.2·43-s + 131.·45-s − 281.·47-s + 405.·49-s + 48.2·51-s + 647.·53-s + 101.·55-s + ⋯ |
| L(s) = 1 | + 1.56·3-s + 0.299·5-s − 1.47·7-s + 1.45·9-s + 0.833·11-s − 0.139·13-s + 0.469·15-s + 0.0845·17-s − 1.29·19-s − 2.31·21-s + 0.208·23-s − 0.910·25-s + 0.719·27-s − 1.74·29-s − 0.0734·31-s + 1.30·33-s − 0.442·35-s − 1.52·37-s − 0.219·39-s − 0.307·41-s − 0.0858·43-s + 0.437·45-s − 0.873·47-s + 1.18·49-s + 0.132·51-s + 1.67·53-s + 0.249·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 8.14T + 27T^{2} \) |
| 5 | \( 1 - 3.35T + 125T^{2} \) |
| 7 | \( 1 + 27.3T + 343T^{2} \) |
| 11 | \( 1 - 30.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 6.55T + 2.19e3T^{2} \) |
| 17 | \( 1 - 5.92T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 272.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 12.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 343.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 80.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 24.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 647.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 263.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 211.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 923.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 645.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 615.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 13.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 359.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 608.T + 9.12e5T^{2} \) |
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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
−8.841375685291259636707657792735, −8.136920021581832172891921438901, −7.06405899561345795341215467103, −6.56182312725476285775133038063, −5.49944846392801787716483752271, −3.89660751907019747589548767170, −3.64265487473079049272223335818, −2.54043078814706097150349040632, −1.73914185863053115155712863254, 0,
1.73914185863053115155712863254, 2.54043078814706097150349040632, 3.64265487473079049272223335818, 3.89660751907019747589548767170, 5.49944846392801787716483752271, 6.56182312725476285775133038063, 7.06405899561345795341215467103, 8.136920021581832172891921438901, 8.841375685291259636707657792735
