| L(s) = 1 | − 0.637·2-s − 2.56·3-s − 1.59·4-s + 5-s + 1.63·6-s − 7-s + 2.29·8-s + 3.59·9-s − 0.637·10-s + 4.09·12-s − 2.09·13-s + 0.637·14-s − 2.56·15-s + 1.72·16-s + 6.47·17-s − 2.29·18-s + 1.61·19-s − 1.59·20-s + 2.56·21-s + 0.929·23-s − 5.88·24-s + 25-s + 1.33·26-s − 1.52·27-s + 1.59·28-s − 2.41·29-s + 1.63·30-s + ⋯ |
| L(s) = 1 | − 0.451·2-s − 1.48·3-s − 0.796·4-s + 0.447·5-s + 0.668·6-s − 0.377·7-s + 0.810·8-s + 1.19·9-s − 0.201·10-s + 1.18·12-s − 0.579·13-s + 0.170·14-s − 0.662·15-s + 0.431·16-s + 1.57·17-s − 0.540·18-s + 0.369·19-s − 0.356·20-s + 0.560·21-s + 0.193·23-s − 1.20·24-s + 0.200·25-s + 0.261·26-s − 0.293·27-s + 0.301·28-s − 0.447·29-s + 0.299·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6203223550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6203223550\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.637T + 2T^{2} \) |
| 3 | \( 1 + 2.56T + 3T^{2} \) |
| 13 | \( 1 + 2.09T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 - 1.61T + 19T^{2} \) |
| 23 | \( 1 - 0.929T + 23T^{2} \) |
| 29 | \( 1 + 2.41T + 29T^{2} \) |
| 31 | \( 1 + 0.749T + 31T^{2} \) |
| 37 | \( 1 - 1.51T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 2.78T + 43T^{2} \) |
| 47 | \( 1 - 0.403T + 47T^{2} \) |
| 53 | \( 1 - 5.79T + 53T^{2} \) |
| 59 | \( 1 - 8.93T + 59T^{2} \) |
| 61 | \( 1 + 8.13T + 61T^{2} \) |
| 67 | \( 1 + 0.789T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 3.89T + 73T^{2} \) |
| 79 | \( 1 + 5.86T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−8.443714715123266493041516733764, −7.51671468180289002557408074245, −7.02757102290435684836327072797, −5.81279724437249517959331384158, −5.66498368436371605570367031470, −4.84595392973163437482729397727, −4.07236619560354025355727151532, −2.94654779255398909215649967950, −1.42369834187236349080823586888, −0.57095863514621023904663267088,
0.57095863514621023904663267088, 1.42369834187236349080823586888, 2.94654779255398909215649967950, 4.07236619560354025355727151532, 4.84595392973163437482729397727, 5.66498368436371605570367031470, 5.81279724437249517959331384158, 7.02757102290435684836327072797, 7.51671468180289002557408074245, 8.443714715123266493041516733764
