| L(s) = 1 | − 1.81·2-s + 3-s + 1.28·4-s − 2.10·5-s − 1.81·6-s + 7-s + 1.28·8-s + 9-s + 3.81·10-s + 1.28·12-s + 0.186·13-s − 1.81·14-s − 2.10·15-s − 4.91·16-s − 2.10·17-s − 1.81·18-s + 4.52·19-s − 2.71·20-s + 21-s − 1.94·23-s + 1.28·24-s − 0.578·25-s − 0.338·26-s + 27-s + 1.28·28-s + 0.186·29-s + 3.81·30-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 0.577·3-s + 0.644·4-s − 0.940·5-s − 0.740·6-s + 0.377·7-s + 0.455·8-s + 0.333·9-s + 1.20·10-s + 0.372·12-s + 0.0516·13-s − 0.484·14-s − 0.542·15-s − 1.22·16-s − 0.509·17-s − 0.427·18-s + 1.03·19-s − 0.606·20-s + 0.218·21-s − 0.405·23-s + 0.263·24-s − 0.115·25-s − 0.0662·26-s + 0.192·27-s + 0.243·28-s + 0.0346·29-s + 0.696·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8723379445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8723379445\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 5 | \( 1 + 2.10T + 5T^{2} \) |
| 13 | \( 1 - 0.186T + 13T^{2} \) |
| 17 | \( 1 + 2.10T + 17T^{2} \) |
| 19 | \( 1 - 4.52T + 19T^{2} \) |
| 23 | \( 1 + 1.94T + 23T^{2} \) |
| 29 | \( 1 - 0.186T + 29T^{2} \) |
| 31 | \( 1 - 4.20T + 31T^{2} \) |
| 37 | \( 1 + 1.28T + 37T^{2} \) |
| 41 | \( 1 - 8.91T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 9.01T + 47T^{2} \) |
| 53 | \( 1 + 6.71T + 53T^{2} \) |
| 59 | \( 1 + 4.44T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 5.07T + 67T^{2} \) |
| 71 | \( 1 - 6.72T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 7.45T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 0.946T + 89T^{2} \) |
| 97 | \( 1 - 7.27T + 97T^{2} \) |
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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
−8.747082457569969443054747866605, −8.252994166482540368843373831833, −7.56375681112644954103959286057, −7.20574887483134686128397654114, −6.00908020744015891525317656594, −4.70514588660611619215474762097, −4.10596705855977111930261894264, −2.99496505010097958401755921947, −1.85923650803579306149926222343, −0.71023939875706387668511275737,
0.71023939875706387668511275737, 1.85923650803579306149926222343, 2.99496505010097958401755921947, 4.10596705855977111930261894264, 4.70514588660611619215474762097, 6.00908020744015891525317656594, 7.20574887483134686128397654114, 7.56375681112644954103959286057, 8.252994166482540368843373831833, 8.747082457569969443054747866605
