L(s) = 1 | + 3-s − 2·5-s − 7-s + 9-s − 13-s − 2·15-s + 4·17-s − 21-s + 6·23-s − 25-s + 27-s − 6·29-s − 8·31-s + 2·35-s − 2·37-s − 39-s − 10·41-s + 12·43-s − 2·45-s − 6·47-s + 49-s + 4·51-s − 14·53-s + 10·59-s + 14·61-s − 63-s + 2·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.277·13-s − 0.516·15-s + 0.970·17-s − 0.218·21-s + 1.25·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.338·35-s − 0.328·37-s − 0.160·39-s − 1.56·41-s + 1.82·43-s − 0.298·45-s − 0.875·47-s + 1/7·49-s + 0.560·51-s − 1.92·53-s + 1.30·59-s + 1.79·61-s − 0.125·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.024720645073071247166158836960, −7.27252718679211889952820214974, −6.89464530761715085056452086549, −5.67582782438624982660472298289, −5.03412718040629533949227900038, −3.89524322617489290032319095900, −3.52034845393411725677362355537, −2.60486327152445744673496984226, −1.41306389292210855793775592870, 0,
1.41306389292210855793775592870, 2.60486327152445744673496984226, 3.52034845393411725677362355537, 3.89524322617489290032319095900, 5.03412718040629533949227900038, 5.67582782438624982660472298289, 6.89464530761715085056452086549, 7.27252718679211889952820214974, 8.024720645073071247166158836960