| L(s) = 1 | − 2.82·3-s − 5-s − 7-s + 5.00·9-s + 11-s + 4.82·13-s + 2.82·15-s − 0.828·17-s + 2.82·21-s + 4·23-s + 25-s − 5.65·27-s − 2·29-s − 6.82·31-s − 2.82·33-s + 35-s + 11.6·37-s − 13.6·39-s + 8.82·41-s − 1.65·43-s − 5.00·45-s + 2.82·47-s + 49-s + 2.34·51-s + 3.65·53-s − 55-s + 6.82·59-s + ⋯ |
| L(s) = 1 | − 1.63·3-s − 0.447·5-s − 0.377·7-s + 1.66·9-s + 0.301·11-s + 1.33·13-s + 0.730·15-s − 0.200·17-s + 0.617·21-s + 0.834·23-s + 0.200·25-s − 1.08·27-s − 0.371·29-s − 1.22·31-s − 0.492·33-s + 0.169·35-s + 1.91·37-s − 2.18·39-s + 1.37·41-s − 0.252·43-s − 0.745·45-s + 0.412·47-s + 0.142·49-s + 0.328·51-s + 0.502·53-s − 0.134·55-s + 0.888·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9210440254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9210440254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 3.65T + 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 - 3.17T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 3.17T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 + 9.31T + 97T^{2} \) |
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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
−7.908123402405193364210459148466, −7.08235557852766573879249711587, −6.60369776744075058562709058375, −5.78861259835636369185549575833, −5.51248926678965835630395708089, −4.34020811622577044259112023816, −3.98819276744448794967835378609, −2.86534155054653473745408298368, −1.41072520765682018409141049351, −0.60185193535486940080022239391,
0.60185193535486940080022239391, 1.41072520765682018409141049351, 2.86534155054653473745408298368, 3.98819276744448794967835378609, 4.34020811622577044259112023816, 5.51248926678965835630395708089, 5.78861259835636369185549575833, 6.60369776744075058562709058375, 7.08235557852766573879249711587, 7.908123402405193364210459148466
