| L(s) = 1 | + 35.1·2-s + 65.6·3-s + 720.·4-s − 1.36e3·5-s + 2.30e3·6-s − 2.40e3·7-s + 7.30e3·8-s − 1.53e4·9-s − 4.79e4·10-s + 5.85e4·11-s + 4.72e4·12-s − 2.85e4·13-s − 8.42e4·14-s − 8.96e4·15-s − 1.12e5·16-s − 2.50e5·17-s − 5.39e5·18-s − 6.25e3·19-s − 9.82e5·20-s − 1.57e5·21-s + 2.05e6·22-s − 2.02e6·23-s + 4.79e5·24-s − 8.97e4·25-s − 1.00e6·26-s − 2.30e6·27-s − 1.72e6·28-s + ⋯ |
| L(s) = 1 | + 1.55·2-s + 0.468·3-s + 1.40·4-s − 0.976·5-s + 0.726·6-s − 0.377·7-s + 0.630·8-s − 0.780·9-s − 1.51·10-s + 1.20·11-s + 0.658·12-s − 0.277·13-s − 0.586·14-s − 0.457·15-s − 0.428·16-s − 0.728·17-s − 1.21·18-s − 0.0110·19-s − 1.37·20-s − 0.176·21-s + 1.87·22-s − 1.50·23-s + 0.295·24-s − 0.0459·25-s − 0.430·26-s − 0.833·27-s − 0.531·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + 2.40e3T \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 - 35.1T + 512T^{2} \) |
| 3 | \( 1 - 65.6T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.36e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 5.85e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 2.50e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 6.25e3T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.02e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.90e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.97e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.94e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 6.96e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.97e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.23e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.80e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.50e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.82e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.75e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.39e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.35e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.57e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 8.60e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.39e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.39e8T + 7.60e17T^{2} \) |
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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
−11.81581721807972020255922288087, −11.36210998515175413856009862454, −9.444330595117748634669164205452, −8.198781644549297467058521091484, −6.82698001597108714217423651665, −5.74970486846810041442961481989, −4.18869544926845547607349007979, −3.60276052316692271947712748548, −2.29579064958615051981537928676, 0,
2.29579064958615051981537928676, 3.60276052316692271947712748548, 4.18869544926845547607349007979, 5.74970486846810041442961481989, 6.82698001597108714217423651665, 8.198781644549297467058521091484, 9.444330595117748634669164205452, 11.36210998515175413856009862454, 11.81581721807972020255922288087
