| L(s) = 1 | − 2.56·3-s + 2·5-s + 0.561·7-s + 3.56·9-s − 11-s + 0.561·13-s − 5.12·15-s − 0.561·17-s + 19-s − 1.43·21-s − 1.43·23-s − 25-s − 1.43·27-s − 5.68·29-s − 2·31-s + 2.56·33-s + 1.12·35-s − 5.12·37-s − 1.43·39-s + 2·41-s + 7.12·45-s − 8·47-s − 6.68·49-s + 1.43·51-s + 12.8·53-s − 2·55-s − 2.56·57-s + ⋯ |
| L(s) = 1 | − 1.47·3-s + 0.894·5-s + 0.212·7-s + 1.18·9-s − 0.301·11-s + 0.155·13-s − 1.32·15-s − 0.136·17-s + 0.229·19-s − 0.313·21-s − 0.299·23-s − 0.200·25-s − 0.276·27-s − 1.05·29-s − 0.359·31-s + 0.445·33-s + 0.189·35-s − 0.842·37-s − 0.230·39-s + 0.312·41-s + 1.06·45-s − 1.16·47-s − 0.954·49-s + 0.201·51-s + 1.75·53-s − 0.269·55-s − 0.339·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 0.561T + 7T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 17 | \( 1 + 0.561T + 17T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 - 7.68T + 59T^{2} \) |
| 61 | \( 1 - 6.24T + 61T^{2} \) |
| 67 | \( 1 - 7.68T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 9.68T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 0.876T + 89T^{2} \) |
| 97 | \( 1 + 7.12T + 97T^{2} \) |
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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
−8.241358901127949633502169989892, −7.24090246049803484143716296167, −6.62556465392806951888721575258, −5.72154990308849569754662416385, −5.51584129003431748766826359571, −4.66891664935232374875785439530, −3.65445867279053163918845891013, −2.29194267295121904310005997608, −1.34577154348212522773253920612, 0,
1.34577154348212522773253920612, 2.29194267295121904310005997608, 3.65445867279053163918845891013, 4.66891664935232374875785439530, 5.51584129003431748766826359571, 5.72154990308849569754662416385, 6.62556465392806951888721575258, 7.24090246049803484143716296167, 8.241358901127949633502169989892
