| L(s) = 1 | + 2.87·3-s + 2.47·5-s − 7-s + 5.24·9-s + 2.87·11-s + 13-s + 7.11·15-s + 0.624·17-s + 0.478·19-s − 2.87·21-s − 3.35·23-s + 1.14·25-s + 6.45·27-s − 4.64·29-s − 9.51·31-s + 8.24·33-s − 2.47·35-s + 4.87·37-s + 2.87·39-s + 1.28·41-s − 1.85·43-s + 13.0·45-s − 9.51·47-s + 49-s + 1.79·51-s + 8.64·53-s + 7.11·55-s + ⋯ |
| L(s) = 1 | + 1.65·3-s + 1.10·5-s − 0.377·7-s + 1.74·9-s + 0.865·11-s + 0.277·13-s + 1.83·15-s + 0.151·17-s + 0.109·19-s − 0.626·21-s − 0.698·23-s + 0.229·25-s + 1.24·27-s − 0.861·29-s − 1.70·31-s + 1.43·33-s − 0.419·35-s + 0.800·37-s + 0.459·39-s + 0.201·41-s − 0.282·43-s + 1.93·45-s − 1.38·47-s + 0.142·49-s + 0.251·51-s + 1.18·53-s + 0.959·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.653284264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.653284264\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.87T + 3T^{2} \) |
| 5 | \( 1 - 2.47T + 5T^{2} \) |
| 11 | \( 1 - 2.87T + 11T^{2} \) |
| 17 | \( 1 - 0.624T + 17T^{2} \) |
| 19 | \( 1 - 0.478T + 19T^{2} \) |
| 23 | \( 1 + 3.35T + 23T^{2} \) |
| 29 | \( 1 + 4.64T + 29T^{2} \) |
| 31 | \( 1 + 9.51T + 31T^{2} \) |
| 37 | \( 1 - 4.87T + 37T^{2} \) |
| 41 | \( 1 - 1.28T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 + 9.51T + 47T^{2} \) |
| 53 | \( 1 - 8.64T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 9.99T + 67T^{2} \) |
| 71 | \( 1 + 4.36T + 71T^{2} \) |
| 73 | \( 1 + 2.72T + 73T^{2} \) |
| 79 | \( 1 + 0.136T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−9.522293226393407066096205732234, −8.883358482682676317032156270610, −8.061070322261674738544552474760, −7.20904845634999987498856478263, −6.33879698388336291564922862033, −5.46022103138563030282404852668, −4.04750067369301721186081708219, −3.41608767134629788398997252777, −2.30462699691378342870096063516, −1.58139965699532698646216061791,
1.58139965699532698646216061791, 2.30462699691378342870096063516, 3.41608767134629788398997252777, 4.04750067369301721186081708219, 5.46022103138563030282404852668, 6.33879698388336291564922862033, 7.20904845634999987498856478263, 8.061070322261674738544552474760, 8.883358482682676317032156270610, 9.522293226393407066096205732234
