| L(s) = 1 | − 2.95·3-s + 5-s + 3.95·7-s + 5.74·9-s − 0.957·11-s − 2.74·13-s − 2.95·15-s + 5.74·17-s − 6.74·19-s − 11.7·21-s + 23-s + 25-s − 8.12·27-s + 5.21·29-s − 5.95·31-s + 2.83·33-s + 3.95·35-s + 9.12·37-s + 8.12·39-s + 0.252·41-s + 8·43-s + 5.74·45-s + 5.49·47-s + 8.66·49-s − 16.9·51-s − 7.12·53-s − 0.957·55-s + ⋯ |
| L(s) = 1 | − 1.70·3-s + 0.447·5-s + 1.49·7-s + 1.91·9-s − 0.288·11-s − 0.761·13-s − 0.763·15-s + 1.39·17-s − 1.54·19-s − 2.55·21-s + 0.208·23-s + 0.200·25-s − 1.56·27-s + 0.967·29-s − 1.07·31-s + 0.493·33-s + 0.668·35-s + 1.50·37-s + 1.30·39-s + 0.0394·41-s + 1.21·43-s + 0.856·45-s + 0.801·47-s + 1.23·49-s − 2.38·51-s − 0.978·53-s − 0.129·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.099476904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.099476904\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2.95T + 3T^{2} \) |
| 7 | \( 1 - 3.95T + 7T^{2} \) |
| 11 | \( 1 + 0.957T + 11T^{2} \) |
| 13 | \( 1 + 2.74T + 13T^{2} \) |
| 17 | \( 1 - 5.74T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 29 | \( 1 - 5.21T + 29T^{2} \) |
| 31 | \( 1 + 5.95T + 31T^{2} \) |
| 37 | \( 1 - 9.12T + 37T^{2} \) |
| 41 | \( 1 - 0.252T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 5.49T + 47T^{2} \) |
| 53 | \( 1 + 7.12T + 53T^{2} \) |
| 59 | \( 1 + 4.78T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 9.12T + 67T^{2} \) |
| 71 | \( 1 - 1.66T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 0.704T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−10.39278810627372267625194181510, −9.516521648956374179341573244447, −8.210824853956913487815980141276, −7.46680651700850437000737249647, −6.45344847258585934584209080982, −5.58319440601672552522792286112, −5.01946537443459750564044220193, −4.24402798863863742446540885582, −2.21276668962332384358640616607, −0.947027120518869112509197893180,
0.947027120518869112509197893180, 2.21276668962332384358640616607, 4.24402798863863742446540885582, 5.01946537443459750564044220193, 5.58319440601672552522792286112, 6.45344847258585934584209080982, 7.46680651700850437000737249647, 8.210824853956913487815980141276, 9.516521648956374179341573244447, 10.39278810627372267625194181510
