| L(s) = 1 | + 6.56e3·3-s − 1.60e6·5-s + 9.41e6·7-s + 4.30e7·9-s + 1.86e8·11-s − 2.62e9·13-s − 1.05e10·15-s + 4.37e10·17-s + 9.65e10·19-s + 6.17e10·21-s − 2.90e11·23-s + 1.82e12·25-s + 2.82e11·27-s + 1.39e12·29-s − 7.64e12·31-s + 1.22e12·33-s − 1.51e13·35-s − 3.33e13·37-s − 1.72e13·39-s − 1.20e13·41-s + 7.55e11·43-s − 6.92e13·45-s + 2.80e14·47-s − 1.43e14·49-s + 2.87e14·51-s + 4.60e14·53-s − 3.00e14·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.84·5-s + 0.617·7-s + 1/3·9-s + 0.262·11-s − 0.892·13-s − 1.06·15-s + 1.52·17-s + 1.30·19-s + 0.356·21-s − 0.774·23-s + 2.39·25-s + 0.192·27-s + 0.519·29-s − 1.61·31-s + 0.151·33-s − 1.13·35-s − 1.56·37-s − 0.515·39-s − 0.235·41-s + 0.00985·43-s − 0.614·45-s + 1.71·47-s − 0.618·49-s + 0.878·51-s + 1.01·53-s − 0.484·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{8} T \) |
| good | 5 | \( 1 + 321786 p T + p^{17} T^{2} \) |
| 7 | \( 1 - 1345312 p T + p^{17} T^{2} \) |
| 11 | \( 1 - 186910524 T + p^{17} T^{2} \) |
| 13 | \( 1 + 201957130 p T + p^{17} T^{2} \) |
| 17 | \( 1 - 43782311106 T + p^{17} T^{2} \) |
| 19 | \( 1 - 96594985540 T + p^{17} T^{2} \) |
| 23 | \( 1 + 290867937336 T + p^{17} T^{2} \) |
| 29 | \( 1 - 1398617429094 T + p^{17} T^{2} \) |
| 31 | \( 1 + 7647898359464 T + p^{17} T^{2} \) |
| 37 | \( 1 + 33369516616762 T + p^{17} T^{2} \) |
| 41 | \( 1 + 12032733393990 T + p^{17} T^{2} \) |
| 43 | \( 1 - 755092495804 T + p^{17} T^{2} \) |
| 47 | \( 1 - 280540358127936 T + p^{17} T^{2} \) |
| 53 | \( 1 - 460570203615582 T + p^{17} T^{2} \) |
| 59 | \( 1 + 1078467799153284 T + p^{17} T^{2} \) |
| 61 | \( 1 + 1980778975313218 T + p^{17} T^{2} \) |
| 67 | \( 1 + 4850190377589884 T + p^{17} T^{2} \) |
| 71 | \( 1 + 2707574704052040 T + p^{17} T^{2} \) |
| 73 | \( 1 + 5002264428090742 T + p^{17} T^{2} \) |
| 79 | \( 1 - 9774477292907752 T + p^{17} T^{2} \) |
| 83 | \( 1 + 17112919183614396 T + p^{17} T^{2} \) |
| 89 | \( 1 - 34698182155846650 T + p^{17} T^{2} \) |
| 97 | \( 1 - 68616916871806082 T + p^{17} T^{2} \) |
| 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
−11.86748055570333740587164030541, −10.44476324236076747485560709096, −8.968290208713404605979544823936, −7.72100828977867917860473214603, −7.40602227710693771101726200797, −5.16775708945650549959086176840, −3.95631985962203533207798548244, −3.08167107943628329465035681638, −1.32709273676141873712957430327, 0,
1.32709273676141873712957430327, 3.08167107943628329465035681638, 3.95631985962203533207798548244, 5.16775708945650549959086176840, 7.40602227710693771101726200797, 7.72100828977867917860473214603, 8.968290208713404605979544823936, 10.44476324236076747485560709096, 11.86748055570333740587164030541
