| L(s) = 1 | + 4.47·7-s − 2.23·11-s + 4·13-s + 7·17-s − 6.70·19-s + 4.47·23-s + 4.47·31-s + 2·37-s − 5·41-s + 8.94·47-s + 13.0·49-s − 6·53-s − 8.94·59-s + 10·61-s − 2.23·67-s + 8.94·71-s − 9·73-s − 10.0·77-s − 4.47·79-s + 11.1·83-s + 5·89-s + 17.8·91-s + 2·97-s + 2·101-s − 8.94·103-s − 2.23·107-s + 6·109-s + ⋯ |
| L(s) = 1 | + 1.69·7-s − 0.674·11-s + 1.10·13-s + 1.69·17-s − 1.53·19-s + 0.932·23-s + 0.803·31-s + 0.328·37-s − 0.780·41-s + 1.30·47-s + 1.85·49-s − 0.824·53-s − 1.16·59-s + 1.28·61-s − 0.273·67-s + 1.06·71-s − 1.05·73-s − 1.13·77-s − 0.503·79-s + 1.22·83-s + 0.529·89-s + 1.87·91-s + 0.203·97-s + 0.199·101-s − 0.881·103-s − 0.216·107-s + 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.850079390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.850079390\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 7T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 2.23T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 5T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.016414042951017515425549788020, −7.44457513384647855913606507315, −6.47358021143222230821054365599, −5.71996527623471335970229643032, −5.08552351767100810682212349067, −4.45406836765585596943349957592, −3.61808640989923263578556035214, −2.64628316044473324759411163793, −1.68893617541577686909930874664, −0.927964214561406531269243865956,
0.927964214561406531269243865956, 1.68893617541577686909930874664, 2.64628316044473324759411163793, 3.61808640989923263578556035214, 4.45406836765585596943349957592, 5.08552351767100810682212349067, 5.71996527623471335970229643032, 6.47358021143222230821054365599, 7.44457513384647855913606507315, 8.016414042951017515425549788020
