| L(s) = 1 | − 124.·2-s − 983.·3-s + 7.23e3·4-s − 8.00e3·5-s + 1.22e5·6-s − 1.52e5·7-s + 1.19e5·8-s − 6.26e5·9-s + 9.93e5·10-s − 5.70e6·11-s − 7.11e6·12-s − 2.81e7·13-s + 1.88e7·14-s + 7.86e6·15-s − 7.40e7·16-s − 1.64e8·17-s + 7.78e7·18-s − 3.03e7·19-s − 5.78e7·20-s + 1.49e8·21-s + 7.08e8·22-s + 1.48e8·23-s − 1.17e8·24-s − 1.15e9·25-s + 3.50e9·26-s + 2.18e9·27-s − 1.09e9·28-s + ⋯ |
| L(s) = 1 | − 1.37·2-s − 0.778·3-s + 0.883·4-s − 0.229·5-s + 1.06·6-s − 0.488·7-s + 0.160·8-s − 0.393·9-s + 0.314·10-s − 0.971·11-s − 0.687·12-s − 1.61·13-s + 0.670·14-s + 0.178·15-s − 1.10·16-s − 1.65·17-s + 0.539·18-s − 0.148·19-s − 0.202·20-s + 0.380·21-s + 1.33·22-s + 0.208·23-s − 0.125·24-s − 0.947·25-s + 2.22·26-s + 1.08·27-s − 0.431·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.03275233893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03275233893\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 1.48e8T \) |
| good | 2 | \( 1 + 124.T + 8.19e3T^{2} \) |
| 3 | \( 1 + 983.T + 1.59e6T^{2} \) |
| 5 | \( 1 + 8.00e3T + 1.22e9T^{2} \) |
| 7 | \( 1 + 1.52e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 5.70e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 2.81e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.64e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.03e7T + 4.20e16T^{2} \) |
| 29 | \( 1 + 2.12e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 2.12e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.36e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 4.13e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 3.33e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 3.60e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.32e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 3.46e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 3.98e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 3.67e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 8.01e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 2.09e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 3.14e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.99e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 1.20e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 2.87e12T + 6.73e25T^{2} \) |
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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
−15.27786861327931316687011326514, −13.27228164927238671079353694710, −11.71977761636560152398560552181, −10.60736909881492983718345492398, −9.492285384273262227551648428410, −8.058846028375649362924759557429, −6.73357249097214038260232911859, −4.91481662636718548984643255493, −2.34075968529421836067800603712, −0.14063279881094978765074088446,
0.14063279881094978765074088446, 2.34075968529421836067800603712, 4.91481662636718548984643255493, 6.73357249097214038260232911859, 8.058846028375649362924759557429, 9.492285384273262227551648428410, 10.60736909881492983718345492398, 11.71977761636560152398560552181, 13.27228164927238671079353694710, 15.27786861327931316687011326514
