L(s) = 1 | + 3-s − 5-s − 2.82·7-s + 9-s + 2.82·11-s − 4.82·13-s − 15-s + 3.65·17-s + 19-s − 2.82·21-s − 4·23-s + 25-s + 27-s + 4.82·29-s + 2.82·33-s + 2.82·35-s + 0.828·37-s − 4.82·39-s + 4.82·41-s + 2.82·43-s − 45-s − 4·47-s + 1.00·49-s + 3.65·51-s − 2·53-s − 2.82·55-s + 57-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.06·7-s + 0.333·9-s + 0.852·11-s − 1.33·13-s − 0.258·15-s + 0.886·17-s + 0.229·19-s − 0.617·21-s − 0.834·23-s + 0.200·25-s + 0.192·27-s + 0.896·29-s + 0.492·33-s + 0.478·35-s + 0.136·37-s − 0.773·39-s + 0.754·41-s + 0.431·43-s − 0.149·45-s − 0.583·47-s + 0.142·49-s + 0.512·51-s − 0.274·53-s − 0.381·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 0.828T + 37T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 9.65T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 0.828T + 97T^{2} \) |
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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
−7.71771676946841932791460204748, −7.49314161444815504826321043461, −6.52827737111957505618378066263, −5.95867221759944079146496900026, −4.80139748531886841141519002397, −4.11538801448683417180965887870, −3.23250443894761546937962744477, −2.69757400236581797602947353197, −1.40301965935946598728794464790, 0,
1.40301965935946598728794464790, 2.69757400236581797602947353197, 3.23250443894761546937962744477, 4.11538801448683417180965887870, 4.80139748531886841141519002397, 5.95867221759944079146496900026, 6.52827737111957505618378066263, 7.49314161444815504826321043461, 7.71771676946841932791460204748