| L(s) = 1 | − 3-s − 5-s + 3.23·7-s + 9-s − 3.23·11-s + 1.23·13-s + 15-s − 4.47·17-s − 19-s − 3.23·21-s − 2.47·23-s + 25-s − 27-s − 2.76·29-s + 4·31-s + 3.23·33-s − 3.23·35-s − 5.23·37-s − 1.23·39-s + 3.70·41-s + 3.23·43-s − 45-s − 2.47·47-s + 3.47·49-s + 4.47·51-s − 8.47·53-s + 3.23·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.22·7-s + 0.333·9-s − 0.975·11-s + 0.342·13-s + 0.258·15-s − 1.08·17-s − 0.229·19-s − 0.706·21-s − 0.515·23-s + 0.200·25-s − 0.192·27-s − 0.513·29-s + 0.718·31-s + 0.563·33-s − 0.546·35-s − 0.860·37-s − 0.197·39-s + 0.579·41-s + 0.493·43-s − 0.149·45-s − 0.360·47-s + 0.496·49-s + 0.626·51-s − 1.16·53-s + 0.436·55-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 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 5.23T + 37T^{2} \) |
| 41 | \( 1 - 3.70T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 - 2.47T + 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 - 8.47T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 4.29T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−8.379143774009457980965854057029, −7.978415956055006909622991924822, −7.13878928477746970218011266741, −6.25619396920200519612960480981, −5.33423631946843268811976044541, −4.68785295631083269829388828096, −3.95775263632380092484399705399, −2.58783422159317136510033702862, −1.52174203513608166529632919729, 0,
1.52174203513608166529632919729, 2.58783422159317136510033702862, 3.95775263632380092484399705399, 4.68785295631083269829388828096, 5.33423631946843268811976044541, 6.25619396920200519612960480981, 7.13878928477746970218011266741, 7.978415956055006909622991924822, 8.379143774009457980965854057029
