| L(s) = 1 | − 1.05·2-s + 5.26·3-s − 30.8·4-s − 25·5-s − 5.53·6-s − 191.·7-s + 66.0·8-s − 215.·9-s + 26.2·10-s − 121·11-s − 162.·12-s − 294.·13-s + 201.·14-s − 131.·15-s + 919.·16-s + 179.·17-s + 226.·18-s − 361·19-s + 772.·20-s − 1.00e3·21-s + 127.·22-s + 3.38e3·23-s + 348.·24-s + 625·25-s + 309.·26-s − 2.41e3·27-s + 5.91e3·28-s + ⋯ |
| L(s) = 1 | − 0.185·2-s + 0.337·3-s − 0.965·4-s − 0.447·5-s − 0.0627·6-s − 1.47·7-s + 0.365·8-s − 0.885·9-s + 0.0830·10-s − 0.301·11-s − 0.326·12-s − 0.482·13-s + 0.274·14-s − 0.151·15-s + 0.897·16-s + 0.150·17-s + 0.164·18-s − 0.229·19-s + 0.431·20-s − 0.499·21-s + 0.0560·22-s + 1.33·23-s + 0.123·24-s + 0.200·25-s + 0.0896·26-s − 0.637·27-s + 1.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 1.05T + 32T^{2} \) |
| 3 | \( 1 - 5.26T + 243T^{2} \) |
| 7 | \( 1 + 191.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 294.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 179.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.38e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 852.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.31e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.13e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.73e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.21e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.73e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.06e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.49e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.74e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.42e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.63e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.85e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.74e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.71e5T + 8.58e9T^{2} \) |
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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
−8.786718350473030833441863799635, −8.164996344218953669297108852086, −7.21448653471827308319100476403, −6.23355914981654124815469302665, −5.28396275642439517441514969445, −4.30485019822873995427081095843, −3.28350653070660274841791366423, −2.69776270541399690220739996183, −0.801909334095125053779341696859, 0,
0.801909334095125053779341696859, 2.69776270541399690220739996183, 3.28350653070660274841791366423, 4.30485019822873995427081095843, 5.28396275642439517441514969445, 6.23355914981654124815469302665, 7.21448653471827308319100476403, 8.164996344218953669297108852086, 8.786718350473030833441863799635
