| L(s) = 1 | − 3-s − 5-s − 1.23·7-s + 9-s + 1.23·11-s − 3.23·13-s + 15-s + 4.47·17-s − 19-s + 1.23·21-s + 6.47·23-s + 25-s − 27-s − 7.23·29-s + 4·31-s − 1.23·33-s + 1.23·35-s − 0.763·37-s + 3.23·39-s − 9.70·41-s − 1.23·43-s − 45-s + 6.47·47-s − 5.47·49-s − 4.47·51-s + 0.472·53-s − 1.23·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.467·7-s + 0.333·9-s + 0.372·11-s − 0.897·13-s + 0.258·15-s + 1.08·17-s − 0.229·19-s + 0.269·21-s + 1.34·23-s + 0.200·25-s − 0.192·27-s − 1.34·29-s + 0.718·31-s − 0.215·33-s + 0.208·35-s − 0.125·37-s + 0.518·39-s − 1.51·41-s − 0.188·43-s − 0.149·45-s + 0.944·47-s − 0.781·49-s − 0.626·51-s + 0.0648·53-s − 0.166·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 7.23T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 0.763T + 37T^{2} \) |
| 41 | \( 1 + 9.70T + 41T^{2} \) |
| 43 | \( 1 + 1.23T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 6.47T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 0.472T + 73T^{2} \) |
| 79 | \( 1 - 4.94T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 - 4.18T + 97T^{2} \) |
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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
−8.691285179553511532105644024016, −7.65569917793388420729939516689, −7.12213751654236676472690766277, −6.32877326432388639295631973237, −5.41295118083384829701574775311, −4.71991735134997732513314414930, −3.70103776385555526641296295645, −2.85373370141255988126225011754, −1.38337987434094912643063509065, 0,
1.38337987434094912643063509065, 2.85373370141255988126225011754, 3.70103776385555526641296295645, 4.71991735134997732513314414930, 5.41295118083384829701574775311, 6.32877326432388639295631973237, 7.12213751654236676472690766277, 7.65569917793388420729939516689, 8.691285179553511532105644024016
