L(s) = 1 | − 0.146·3-s + 3.56·5-s − 2.97·9-s − 5.58·11-s − 6.25·13-s − 0.522·15-s − 6.24·17-s + 7.23·19-s + 5.27·23-s + 7.73·25-s + 0.875·27-s − 4.42·29-s + 1.06·31-s + 0.817·33-s + 2.40·37-s + 0.917·39-s − 41-s + 12.2·43-s − 10.6·45-s − 4.35·47-s + 0.915·51-s + 9.39·53-s − 19.9·55-s − 1.05·57-s − 13.2·59-s + 14.3·61-s − 22.3·65-s + ⋯ |
L(s) = 1 | − 0.0845·3-s + 1.59·5-s − 0.992·9-s − 1.68·11-s − 1.73·13-s − 0.135·15-s − 1.51·17-s + 1.65·19-s + 1.10·23-s + 1.54·25-s + 0.168·27-s − 0.821·29-s + 0.190·31-s + 0.142·33-s + 0.395·37-s + 0.146·39-s − 0.156·41-s + 1.87·43-s − 1.58·45-s − 0.635·47-s + 0.128·51-s + 1.29·53-s − 2.68·55-s − 0.140·57-s − 1.73·59-s + 1.83·61-s − 2.77·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.715894842\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.715894842\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + 0.146T + 3T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 11 | \( 1 + 5.58T + 11T^{2} \) |
| 13 | \( 1 + 6.25T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 - 5.27T + 23T^{2} \) |
| 29 | \( 1 + 4.42T + 29T^{2} \) |
| 31 | \( 1 - 1.06T + 31T^{2} \) |
| 37 | \( 1 - 2.40T + 37T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 4.35T + 47T^{2} \) |
| 53 | \( 1 - 9.39T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 + 4.58T + 67T^{2} \) |
| 71 | \( 1 + 0.0228T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 1.46T + 79T^{2} \) |
| 83 | \( 1 - 4.21T + 83T^{2} \) |
| 89 | \( 1 + 3.56T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−7.65377180920620246934328554999, −7.22544862728090793326816064901, −6.33891596302945409226456512795, −5.52803757026844189912906070211, −5.23837424791495890079491038546, −4.68424382782600430657916523229, −3.11484608489819833308808827615, −2.49503686934105726620473334755, −2.15268724227087274290985271863, −0.60855407371942997857355931442,
0.60855407371942997857355931442, 2.15268724227087274290985271863, 2.49503686934105726620473334755, 3.11484608489819833308808827615, 4.68424382782600430657916523229, 5.23837424791495890079491038546, 5.52803757026844189912906070211, 6.33891596302945409226456512795, 7.22544862728090793326816064901, 7.65377180920620246934328554999