| L(s) = 1 | + 6.31·2-s + 11.8·3-s + 7.82·4-s + 25·5-s + 74.6·6-s − 152.·8-s − 103.·9-s + 157.·10-s − 440.·11-s + 92.5·12-s − 878.·13-s + 295.·15-s − 1.21e3·16-s − 604.·17-s − 650.·18-s − 1.18e3·19-s + 195.·20-s − 2.77e3·22-s + 3.60e3·23-s − 1.80e3·24-s + 625·25-s − 5.54e3·26-s − 4.09e3·27-s − 1.38e3·29-s + 1.86e3·30-s + 8.36e3·31-s − 2.77e3·32-s + ⋯ |
| L(s) = 1 | + 1.11·2-s + 0.758·3-s + 0.244·4-s + 0.447·5-s + 0.846·6-s − 0.842·8-s − 0.424·9-s + 0.498·10-s − 1.09·11-s + 0.185·12-s − 1.44·13-s + 0.339·15-s − 1.18·16-s − 0.507·17-s − 0.473·18-s − 0.752·19-s + 0.109·20-s − 1.22·22-s + 1.42·23-s − 0.639·24-s + 0.200·25-s − 1.60·26-s − 1.08·27-s − 0.305·29-s + 0.378·30-s + 1.56·31-s − 0.478·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 6.31T + 32T^{2} \) |
| 3 | \( 1 - 11.8T + 243T^{2} \) |
| 11 | \( 1 + 440.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 878.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 604.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.18e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.38e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.36e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.10e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.06e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.18e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 431.T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.77e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.42e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.91e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.35e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.94e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.14e5T + 8.58e9T^{2} \) |
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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
−10.86314866504572774351076374463, −9.667947487953086531555172859505, −8.842842576124065836635368497495, −7.77515458093949370966079692081, −6.46734228772654775299634616545, −5.29513348722079222086837869251, −4.53878721045687110868303956481, −2.97674850770892017166736544483, −2.42790238053366169837696624005, 0,
2.42790238053366169837696624005, 2.97674850770892017166736544483, 4.53878721045687110868303956481, 5.29513348722079222086837869251, 6.46734228772654775299634616545, 7.77515458093949370966079692081, 8.842842576124065836635368497495, 9.667947487953086531555172859505, 10.86314866504572774351076374463
