L(s) = 1 | − 2-s + 1.96·3-s + 4-s + 0.891·5-s − 1.96·6-s − 8-s + 2.86·9-s − 0.891·10-s − 0.184·11-s + 1.96·12-s − 1.70·13-s + 1.75·15-s + 16-s + 1.47·17-s − 2.86·18-s + 0.891·20-s + 0.184·22-s − 1.47·23-s − 1.96·24-s − 0.205·25-s + 1.70·26-s + 3.66·27-s − 0.547·29-s − 1.75·30-s − 32-s − 0.362·33-s − 1.47·34-s + ⋯ |
L(s) = 1 | − 2-s + 1.96·3-s + 4-s + 0.891·5-s − 1.96·6-s − 8-s + 2.86·9-s − 0.891·10-s − 0.184·11-s + 1.96·12-s − 1.70·13-s + 1.75·15-s + 16-s + 1.47·17-s − 2.86·18-s + 0.891·20-s + 0.184·22-s − 1.47·23-s − 1.96·24-s − 0.205·25-s + 1.70·26-s + 3.66·27-s − 0.547·29-s − 1.75·30-s − 32-s − 0.362·33-s − 1.47·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.803399702\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.803399702\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 701 | \( 1 - T \) |
good | 3 | \( 1 - 1.96T + T^{2} \) |
| 5 | \( 1 - 0.891T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 0.184T + T^{2} \) |
| 13 | \( 1 + 1.70T + T^{2} \) |
| 17 | \( 1 - 1.47T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.47T + T^{2} \) |
| 29 | \( 1 + 0.547T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.20T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 0.547T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.86T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 0.184T + T^{2} \) |
| 71 | \( 1 + 0.891T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.20T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.86T + T^{2} \) |
| 97 | \( 1 + 0.547T + T^{2} \) |
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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
−9.164223316512426229917503993453, −8.191763511248575960296764860182, −7.67504342998833618112762839671, −7.25728708780562904881938754114, −6.18407517561030601991955026244, −5.13916586318834356052499221329, −3.84471633896647768789171412240, −2.93091165829713083256841943632, −2.24302655909573327290725459407, −1.59727607229403092339675820149,
1.59727607229403092339675820149, 2.24302655909573327290725459407, 2.93091165829713083256841943632, 3.84471633896647768789171412240, 5.13916586318834356052499221329, 6.18407517561030601991955026244, 7.25728708780562904881938754114, 7.67504342998833618112762839671, 8.191763511248575960296764860182, 9.164223316512426229917503993453