L(s) = 1 | + 42.9·2-s − 45.1·3-s + 1.32e3·4-s − 608.·5-s − 1.93e3·6-s + 521.·7-s + 3.50e4·8-s − 1.76e4·9-s − 2.61e4·10-s − 6.18e4·11-s − 5.99e4·12-s + 1.57e5·13-s + 2.23e4·14-s + 2.74e4·15-s + 8.22e5·16-s − 7.57e5·18-s − 3.27e5·19-s − 8.08e5·20-s − 2.35e4·21-s − 2.65e6·22-s − 1.00e6·23-s − 1.58e6·24-s − 1.58e6·25-s + 6.77e6·26-s + 1.68e6·27-s + 6.92e5·28-s − 6.67e6·29-s + ⋯ |
L(s) = 1 | + 1.89·2-s − 0.321·3-s + 2.59·4-s − 0.435·5-s − 0.609·6-s + 0.0820·7-s + 3.02·8-s − 0.896·9-s − 0.825·10-s − 1.27·11-s − 0.834·12-s + 1.53·13-s + 0.155·14-s + 0.140·15-s + 3.13·16-s − 1.70·18-s − 0.575·19-s − 1.13·20-s − 0.0263·21-s − 2.41·22-s − 0.747·23-s − 0.972·24-s − 0.810·25-s + 2.90·26-s + 0.609·27-s + 0.212·28-s − 1.75·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 - 42.9T + 512T^{2} \) |
| 3 | \( 1 + 45.1T + 1.96e4T^{2} \) |
| 5 | \( 1 + 608.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 521.T + 4.03e7T^{2} \) |
| 11 | \( 1 + 6.18e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.57e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 3.27e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.00e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.67e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.67e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 6.88e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 4.90e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.42e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.28e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.52e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 2.13e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.97e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.13e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.17e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 8.46e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.63e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.63e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.16e8T + 7.60e17T^{2} \) |
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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
−10.38487375757171188428633201836, −8.423129739957275618280388470351, −7.58173471497825340766613992619, −6.22230873404470132649140038657, −5.77553278462347181929391003030, −4.75531903546162487097340672707, −3.74986119336170801471830561289, −2.90819304262165734072207494374, −1.76153493959757121858203866161, 0,
1.76153493959757121858203866161, 2.90819304262165734072207494374, 3.74986119336170801471830561289, 4.75531903546162487097340672707, 5.77553278462347181929391003030, 6.22230873404470132649140038657, 7.58173471497825340766613992619, 8.423129739957275618280388470351, 10.38487375757171188428633201836