| L(s) = 1 | + 18.4·2-s − 346.·3-s − 683.·4-s − 6.39e3·6-s − 3.15e4·8-s + 6.09e4·9-s + 2.36e5·12-s + 7.38e5·13-s + 1.18e5·16-s + 1.12e6·18-s − 6.43e6·23-s + 1.09e7·24-s + 9.76e6·25-s + 1.36e7·26-s − 6.54e5·27-s + 3.66e6·29-s − 1.39e7·31-s + 3.44e7·32-s − 4.16e7·36-s − 2.55e8·39-s − 1.18e8·41-s − 1.18e8·46-s + 3.20e8·47-s − 4.09e7·48-s + 2.82e8·49-s + 1.80e8·50-s − 5.04e8·52-s + ⋯ |
| L(s) = 1 | + 0.576·2-s − 1.42·3-s − 0.667·4-s − 0.822·6-s − 0.961·8-s + 1.03·9-s + 0.951·12-s + 1.98·13-s + 0.112·16-s + 0.595·18-s − 23-s + 1.37·24-s + 25-s + 1.14·26-s − 0.0456·27-s + 0.178·29-s − 0.485·31-s + 1.02·32-s − 0.688·36-s − 2.83·39-s − 1.02·41-s − 0.576·46-s + 1.39·47-s − 0.160·48-s + 49-s + 0.576·50-s − 1.32·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.105274450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.105274450\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 6.43e6T \) |
| good | 2 | \( 1 - 18.4T + 1.02e3T^{2} \) |
| 3 | \( 1 + 346.T + 5.90e4T^{2} \) |
| 5 | \( 1 - 9.76e6T^{2} \) |
| 7 | \( 1 - 2.82e8T^{2} \) |
| 11 | \( 1 - 2.59e10T^{2} \) |
| 13 | \( 1 - 7.38e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 2.01e12T^{2} \) |
| 19 | \( 1 - 6.13e12T^{2} \) |
| 29 | \( 1 - 3.66e6T + 4.20e14T^{2} \) |
| 31 | \( 1 + 1.39e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 4.80e15T^{2} \) |
| 41 | \( 1 + 1.18e8T + 1.34e16T^{2} \) |
| 43 | \( 1 - 2.16e16T^{2} \) |
| 47 | \( 1 - 3.20e8T + 5.25e16T^{2} \) |
| 53 | \( 1 - 1.74e17T^{2} \) |
| 59 | \( 1 - 1.28e9T + 5.11e17T^{2} \) |
| 61 | \( 1 - 7.13e17T^{2} \) |
| 67 | \( 1 - 1.82e18T^{2} \) |
| 71 | \( 1 - 3.40e8T + 3.25e18T^{2} \) |
| 73 | \( 1 - 3.37e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 9.46e18T^{2} \) |
| 83 | \( 1 - 1.55e19T^{2} \) |
| 89 | \( 1 - 3.11e19T^{2} \) |
| 97 | \( 1 - 7.37e19T^{2} \) |
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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
−15.62141799977032819688428959915, −13.96878250719895267053061405226, −12.83740336747426397482028707032, −11.68528945851903669530772958510, −10.49622864899576354265041174932, −8.685406240319536627819478370885, −6.35793658152054428545423945367, −5.36486523842997153150179766668, −3.88327592118911807946096010260, −0.78798476304352497427027156520,
0.78798476304352497427027156520, 3.88327592118911807946096010260, 5.36486523842997153150179766668, 6.35793658152054428545423945367, 8.685406240319536627819478370885, 10.49622864899576354265041174932, 11.68528945851903669530772958510, 12.83740336747426397482028707032, 13.96878250719895267053061405226, 15.62141799977032819688428959915
