| L(s) = 1 | + 8.77·2-s − 9·3-s + 45.0·4-s − 17.5·5-s − 79.0·6-s + 12.9·7-s + 114.·8-s + 81·9-s − 153.·10-s − 434.·11-s − 405.·12-s − 637.·13-s + 113.·14-s + 157.·15-s − 433.·16-s − 850.·17-s + 711.·18-s + 1.82e3·19-s − 789.·20-s − 116.·21-s − 3.81e3·22-s + 715.·23-s − 1.03e3·24-s − 2.81e3·25-s − 5.59e3·26-s − 729·27-s + 584.·28-s + ⋯ |
| L(s) = 1 | + 1.55·2-s − 0.577·3-s + 1.40·4-s − 0.313·5-s − 0.896·6-s + 0.0999·7-s + 0.634·8-s + 0.333·9-s − 0.486·10-s − 1.08·11-s − 0.813·12-s − 1.04·13-s + 0.155·14-s + 0.180·15-s − 0.423·16-s − 0.713·17-s + 0.517·18-s + 1.15·19-s − 0.441·20-s − 0.0576·21-s − 1.67·22-s + 0.282·23-s − 0.366·24-s − 0.901·25-s − 1.62·26-s − 0.192·27-s + 0.140·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 - 8.77T + 32T^{2} \) |
| 5 | \( 1 + 17.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 12.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 434.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 637.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 850.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.82e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 715.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.13e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 107.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.61e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.99e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.10e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 2.29e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.35e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.65e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.57e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.10e4T + 8.58e9T^{2} \) |
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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
−11.61493548979022311593746718068, −10.78034315170231749179858965665, −9.488692356861375857158177338623, −7.74942466121667149284571639176, −6.81199967031303685497503191057, −5.46076912963869279406233881866, −4.92952099801868788953954305570, −3.64197115376387266212205742914, −2.30085111527069423611777125621, 0,
2.30085111527069423611777125621, 3.64197115376387266212205742914, 4.92952099801868788953954305570, 5.46076912963869279406233881866, 6.81199967031303685497503191057, 7.74942466121667149284571639176, 9.488692356861375857158177338623, 10.78034315170231749179858965665, 11.61493548979022311593746718068
