| L(s) = 1 | − 21.3·2-s + 86.9·3-s + 326.·4-s − 188.·5-s − 1.85e3·6-s + 639.·7-s − 4.21e3·8-s + 5.37e3·9-s + 4.01e3·10-s − 1.62e3·11-s + 2.83e4·12-s + 1.13e3·13-s − 1.36e4·14-s − 1.63e4·15-s + 4.81e4·16-s + 2.87e4·17-s − 1.14e5·18-s + 4.33e4·19-s − 6.13e4·20-s + 5.55e4·21-s + 3.47e4·22-s − 1.21e4·23-s − 3.66e5·24-s − 4.26e4·25-s − 2.42e4·26-s + 2.77e5·27-s + 2.08e5·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 1.85·3-s + 2.54·4-s − 0.673·5-s − 3.50·6-s + 0.704·7-s − 2.91·8-s + 2.45·9-s + 1.26·10-s − 0.369·11-s + 4.73·12-s + 0.143·13-s − 1.32·14-s − 1.25·15-s + 2.94·16-s + 1.41·17-s − 4.62·18-s + 1.45·19-s − 1.71·20-s + 1.30·21-s + 0.695·22-s − 0.208·23-s − 5.41·24-s − 0.546·25-s − 0.270·26-s + 2.70·27-s + 1.79·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.318518265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.318518265\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 + 21.3T + 128T^{2} \) |
| 3 | \( 1 - 86.9T + 2.18e3T^{2} \) |
| 5 | \( 1 + 188.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 639.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.62e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.13e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.87e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.33e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 6.25e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.25e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.77e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.30e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.85e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.93e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.36e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 8.08e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.40e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.05e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.83e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.78e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.20e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.98e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.37e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.57e6T + 8.07e13T^{2} \) |
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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
−16.15821995775842894143313504374, −15.35790106991966895930914167237, −14.14401604960006415928744428096, −11.96388702822136759585299076883, −10.26785853857219203134328582703, −9.165011928292588613190795287007, −7.945361692501585490004942454128, −7.58554788445951761117113362692, −3.11629785224852303842550341662, −1.43375652739866954972433886572,
1.43375652739866954972433886572, 3.11629785224852303842550341662, 7.58554788445951761117113362692, 7.945361692501585490004942454128, 9.165011928292588613190795287007, 10.26785853857219203134328582703, 11.96388702822136759585299076883, 14.14401604960006415928744428096, 15.35790106991966895930914167237, 16.15821995775842894143313504374
