| L(s) = 1 | − 38.3·2-s + 247.·3-s + 958.·4-s − 875.·5-s − 9.47e3·6-s − 7.07e3·7-s − 1.71e4·8-s + 4.13e4·9-s + 3.35e4·10-s + 5.52e4·11-s + 2.36e5·12-s − 1.79e5·13-s + 2.71e5·14-s − 2.16e5·15-s + 1.65e5·16-s − 1.58e6·18-s − 5.65e4·19-s − 8.39e5·20-s − 1.74e6·21-s − 2.12e6·22-s + 2.36e6·23-s − 4.22e6·24-s − 1.18e6·25-s + 6.89e6·26-s + 5.35e6·27-s − 6.77e6·28-s − 1.38e6·29-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.76·3-s + 1.87·4-s − 0.626·5-s − 2.98·6-s − 1.11·7-s − 1.47·8-s + 2.10·9-s + 1.06·10-s + 1.13·11-s + 3.29·12-s − 1.74·13-s + 1.88·14-s − 1.10·15-s + 0.631·16-s − 3.55·18-s − 0.0994·19-s − 1.17·20-s − 1.96·21-s − 1.92·22-s + 1.76·23-s − 2.60·24-s − 0.607·25-s + 2.95·26-s + 1.93·27-s − 2.08·28-s − 0.364·29-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 | \( 1 + 38.3T + 512T^{2} \) |
| 3 | \( 1 - 247.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 875.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 7.07e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.52e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.79e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 5.65e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.36e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.38e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.30e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.73e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.41e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.24e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.13e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.71e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.33e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.58e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.25e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 6.66e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.13e6T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.27e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.63e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.86e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.45e7T + 7.60e17T^{2} \) |
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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
−9.420962284204280275349508088292, −9.078594691266352076694775912577, −7.975021332717585274590252059235, −7.34119892775865699120568776405, −6.66222514181203900744464399512, −4.28231319644359085878324862753, −3.10665117090072926172767232497, −2.40497831593821909154744088300, −1.18556978115202356238490335698, 0,
1.18556978115202356238490335698, 2.40497831593821909154744088300, 3.10665117090072926172767232497, 4.28231319644359085878324862753, 6.66222514181203900744464399512, 7.34119892775865699120568776405, 7.975021332717585274590252059235, 9.078594691266352076694775912577, 9.420962284204280275349508088292
