| L(s) = 1 | − 2·5-s + 3.12·11-s + 0.438·13-s − 5.12·17-s − 3.12·19-s + 23-s − 25-s − 3.56·29-s − 2.43·31-s + 8.24·37-s + 9.80·41-s − 8·43-s + 0.684·47-s − 7·49-s − 2·53-s − 6.24·55-s − 10.2·59-s − 4.24·61-s − 0.876·65-s + 3.12·67-s − 13.5·71-s − 14.6·73-s + 3.12·79-s − 14.2·83-s + 10.2·85-s − 11.3·89-s + 6.24·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.941·11-s + 0.121·13-s − 1.24·17-s − 0.716·19-s + 0.208·23-s − 0.200·25-s − 0.661·29-s − 0.437·31-s + 1.35·37-s + 1.53·41-s − 1.21·43-s + 0.0998·47-s − 49-s − 0.274·53-s − 0.842·55-s − 1.33·59-s − 0.543·61-s − 0.108·65-s + 0.381·67-s − 1.60·71-s − 1.71·73-s + 0.351·79-s − 1.56·83-s + 1.11·85-s − 1.20·89-s + 0.640·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 - 0.438T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 29 | \( 1 + 3.56T + 29T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 - 9.80T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 0.684T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 - 3.12T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−8.976249067349437709764688785485, −8.186422433254707433153706269900, −7.40591581544749298185377153402, −6.59877885594318682548851978249, −5.85165657463478403100488548936, −4.45959795138111280343625868226, −4.10806812399186429025010777766, −2.95805025675360105325132450644, −1.64074797369415812029192485669, 0,
1.64074797369415812029192485669, 2.95805025675360105325132450644, 4.10806812399186429025010777766, 4.45959795138111280343625868226, 5.85165657463478403100488548936, 6.59877885594318682548851978249, 7.40591581544749298185377153402, 8.186422433254707433153706269900, 8.976249067349437709764688785485
