L(s) = 1 | + 4.06·5-s − 1.52·7-s + 3.75·11-s − 0.808·13-s + 1.48·17-s − 5.96·19-s + 8.26·23-s + 11.4·25-s + 5.45·29-s + 8.18·31-s − 6.18·35-s − 5.32·37-s − 5.14·41-s − 5.00·43-s − 3.25·47-s − 4.68·49-s − 7.56·53-s + 15.2·55-s + 14.2·59-s + 2.91·61-s − 3.28·65-s + 15.6·67-s − 0.362·71-s − 12.1·73-s − 5.70·77-s + 11.8·79-s + 2.55·83-s + ⋯ |
L(s) = 1 | + 1.81·5-s − 0.575·7-s + 1.13·11-s − 0.224·13-s + 0.359·17-s − 1.36·19-s + 1.72·23-s + 2.29·25-s + 1.01·29-s + 1.46·31-s − 1.04·35-s − 0.875·37-s − 0.803·41-s − 0.763·43-s − 0.474·47-s − 0.669·49-s − 1.03·53-s + 2.05·55-s + 1.84·59-s + 0.373·61-s − 0.407·65-s + 1.90·67-s − 0.0429·71-s − 1.42·73-s − 0.650·77-s + 1.33·79-s + 0.280·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.088337698\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.088337698\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 - 4.06T + 5T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 - 3.75T + 11T^{2} \) |
| 13 | \( 1 + 0.808T + 13T^{2} \) |
| 17 | \( 1 - 1.48T + 17T^{2} \) |
| 19 | \( 1 + 5.96T + 19T^{2} \) |
| 23 | \( 1 - 8.26T + 23T^{2} \) |
| 29 | \( 1 - 5.45T + 29T^{2} \) |
| 31 | \( 1 - 8.18T + 31T^{2} \) |
| 37 | \( 1 + 5.32T + 37T^{2} \) |
| 41 | \( 1 + 5.14T + 41T^{2} \) |
| 43 | \( 1 + 5.00T + 43T^{2} \) |
| 47 | \( 1 + 3.25T + 47T^{2} \) |
| 53 | \( 1 + 7.56T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 2.91T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 + 0.362T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 2.55T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 6.54T + 97T^{2} \) |
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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
−8.440779399023236427443787988174, −6.90643789311514677586191530854, −6.62273064611877769016384126286, −6.16524144883971400949889110073, −5.19412791690958032801314954081, −4.67637736336687766398743048936, −3.47000459460864332326760163046, −2.70455700422211634082656897020, −1.84280456603011009775255852882, −0.978147470439231979520231035755,
0.978147470439231979520231035755, 1.84280456603011009775255852882, 2.70455700422211634082656897020, 3.47000459460864332326760163046, 4.67637736336687766398743048936, 5.19412791690958032801314954081, 6.16524144883971400949889110073, 6.62273064611877769016384126286, 6.90643789311514677586191530854, 8.440779399023236427443787988174