L(s) = 1 | + 3-s − 2.13·5-s + 9-s + 0.298·11-s + 6.43·13-s − 2.13·15-s − 1.11·17-s − 1.01·19-s + 4.29·23-s − 0.441·25-s + 27-s + 0.422·29-s + 10.6·31-s + 0.298·33-s + 5.65·37-s + 6.43·39-s − 8.32·41-s − 7.09·43-s − 2.13·45-s − 8.11·47-s − 1.11·51-s + 3.44·53-s − 0.637·55-s − 1.01·57-s + 6.46·59-s + 5.83·61-s − 13.7·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.954·5-s + 0.333·9-s + 0.0900·11-s + 1.78·13-s − 0.551·15-s − 0.270·17-s − 0.233·19-s + 0.896·23-s − 0.0883·25-s + 0.192·27-s + 0.0784·29-s + 1.91·31-s + 0.0519·33-s + 0.929·37-s + 1.03·39-s − 1.30·41-s − 1.08·43-s − 0.318·45-s − 1.18·47-s − 0.156·51-s + 0.472·53-s − 0.0859·55-s − 0.135·57-s + 0.841·59-s + 0.747·61-s − 1.70·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.379184801\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.379184801\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + 2.13T + 5T^{2} \) |
| 11 | \( 1 - 0.298T + 11T^{2} \) |
| 13 | \( 1 - 6.43T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 + 1.01T + 19T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 - 0.422T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 + 8.32T + 41T^{2} \) |
| 43 | \( 1 + 7.09T + 43T^{2} \) |
| 47 | \( 1 + 8.11T + 47T^{2} \) |
| 53 | \( 1 - 3.44T + 53T^{2} \) |
| 59 | \( 1 - 6.46T + 59T^{2} \) |
| 61 | \( 1 - 5.83T + 61T^{2} \) |
| 67 | \( 1 - 5.05T + 67T^{2} \) |
| 71 | \( 1 + 1.35T + 71T^{2} \) |
| 73 | \( 1 - 2.85T + 73T^{2} \) |
| 79 | \( 1 + 7.69T + 79T^{2} \) |
| 83 | \( 1 - 6.03T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 0.512T + 97T^{2} \) |
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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
−8.050156450627505842952066969751, −6.87237827383061757400774314249, −6.62774443354001580287834483893, −5.68382283218380540540360355172, −4.72249229505787548416671696609, −4.08783829931642598142165915846, −3.46737702166552516749889541728, −2.82970686470315613499665908019, −1.66311325592417088490708806641, −0.74985174541004993109493342577,
0.74985174541004993109493342577, 1.66311325592417088490708806641, 2.82970686470315613499665908019, 3.46737702166552516749889541728, 4.08783829931642598142165915846, 4.72249229505787548416671696609, 5.68382283218380540540360355172, 6.62774443354001580287834483893, 6.87237827383061757400774314249, 8.050156450627505842952066969751