| L(s) = 1 | + 3·3-s − 8·4-s − 5·5-s − 26.8·7-s + 9·9-s − 24·12-s − 26.8·13-s − 15·15-s + 64·16-s + 53.6·17-s + 80.4·19-s + 40·20-s − 80.4·21-s − 72·23-s + 25·25-s + 27·27-s + 214.·28-s + 295.·29-s − 20·31-s + 134.·35-s − 72·36-s + 214·37-s − 80.4·39-s + 26.8·41-s − 241.·43-s − 45·45-s − 264·47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 1.44·7-s + 0.333·9-s − 0.577·12-s − 0.572·13-s − 0.258·15-s + 16-s + 0.765·17-s + 0.971·19-s + 0.447·20-s − 0.836·21-s − 0.652·23-s + 0.200·25-s + 0.192·27-s + 1.44·28-s + 1.89·29-s − 0.115·31-s + 0.647·35-s − 0.333·36-s + 0.950·37-s − 0.330·39-s + 0.102·41-s − 0.856·43-s − 0.149·45-s − 0.819·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 8T^{2} \) |
| 7 | \( 1 + 26.8T + 343T^{2} \) |
| 13 | \( 1 + 26.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 80.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 72T + 1.21e4T^{2} \) |
| 29 | \( 1 - 295.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 20T + 2.97e4T^{2} \) |
| 37 | \( 1 - 214T + 5.06e4T^{2} \) |
| 41 | \( 1 - 26.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 241.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 264T + 1.03e5T^{2} \) |
| 53 | \( 1 - 78T + 1.48e5T^{2} \) |
| 59 | \( 1 - 480T + 2.05e5T^{2} \) |
| 61 | \( 1 + 107.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 524T + 3.00e5T^{2} \) |
| 71 | \( 1 - 492T + 3.57e5T^{2} \) |
| 73 | \( 1 + 939.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 295.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 590.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.18e3T + 9.12e5T^{2} \) |
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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
−8.505339789761480325570388560840, −7.88609479581138711086273790967, −7.04001468293727886004973629698, −6.12364602889260379701964746438, −5.13268025368723137465758497849, −4.22418549880912832545368315040, −3.38357280892382716259717909198, −2.80075488410104803892435845838, −1.03995354163611040087452610159, 0,
1.03995354163611040087452610159, 2.80075488410104803892435845838, 3.38357280892382716259717909198, 4.22418549880912832545368315040, 5.13268025368723137465758497849, 6.12364602889260379701964746438, 7.04001468293727886004973629698, 7.88609479581138711086273790967, 8.505339789761480325570388560840
