| L(s) = 1 | − 7·2-s + 17·4-s + 25·5-s + 12·7-s + 105·8-s − 175·10-s − 112·11-s − 974·13-s − 84·14-s − 1.27e3·16-s − 2.18e3·17-s + 1.42e3·19-s + 425·20-s + 784·22-s − 3.21e3·23-s + 625·25-s + 6.81e3·26-s + 204·28-s + 4.15e3·29-s − 5.68e3·31-s + 5.59e3·32-s + 1.52e4·34-s + 300·35-s + 6.48e3·37-s − 9.94e3·38-s + 2.62e3·40-s − 5.40e3·41-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.531·4-s + 0.447·5-s + 0.0925·7-s + 0.580·8-s − 0.553·10-s − 0.279·11-s − 1.59·13-s − 0.114·14-s − 1.24·16-s − 1.83·17-s + 0.902·19-s + 0.237·20-s + 0.345·22-s − 1.26·23-s + 1/5·25-s + 1.97·26-s + 0.0491·28-s + 0.916·29-s − 1.06·31-s + 0.965·32-s + 2.26·34-s + 0.0413·35-s + 0.778·37-s − 1.11·38-s + 0.259·40-s − 0.501·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 + 7 T + p^{5} T^{2} \) |
| 7 | \( 1 - 12 T + p^{5} T^{2} \) |
| 11 | \( 1 + 112 T + p^{5} T^{2} \) |
| 13 | \( 1 + 974 T + p^{5} T^{2} \) |
| 17 | \( 1 + 2182 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1420 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3216 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4150 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5688 T + p^{5} T^{2} \) |
| 37 | \( 1 - 6482 T + p^{5} T^{2} \) |
| 41 | \( 1 + 5402 T + p^{5} T^{2} \) |
| 43 | \( 1 + 21764 T + p^{5} T^{2} \) |
| 47 | \( 1 - 368 T + p^{5} T^{2} \) |
| 53 | \( 1 + 12586 T + p^{5} T^{2} \) |
| 59 | \( 1 - 25520 T + p^{5} T^{2} \) |
| 61 | \( 1 - 11782 T + p^{5} T^{2} \) |
| 67 | \( 1 + 13188 T + p^{5} T^{2} \) |
| 71 | \( 1 - 35968 T + p^{5} T^{2} \) |
| 73 | \( 1 - 73186 T + p^{5} T^{2} \) |
| 79 | \( 1 + 52440 T + p^{5} T^{2} \) |
| 83 | \( 1 + 69036 T + p^{5} T^{2} \) |
| 89 | \( 1 - 33870 T + p^{5} T^{2} \) |
| 97 | \( 1 - 143042 T + p^{5} T^{2} \) |
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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
−14.27137294936535674695377748047, −13.10120249854867078543965018590, −11.54278706235968374423247103783, −10.20616925550241895149060627527, −9.416515431562454615825374180429, −8.135960902554524287570762848093, −6.87701619293363042135713139570, −4.84118619701970776334685250643, −2.11016658508599465163700624335, 0,
2.11016658508599465163700624335, 4.84118619701970776334685250643, 6.87701619293363042135713139570, 8.135960902554524287570762848093, 9.416515431562454615825374180429, 10.20616925550241895149060627527, 11.54278706235968374423247103783, 13.10120249854867078543965018590, 14.27137294936535674695377748047
