| L(s) = 1 | + 3-s − 5-s − 3.37·7-s + 9-s − 0.627·11-s − 13-s − 15-s − 5.37·17-s − 6.74·19-s − 3.37·21-s + 7.37·23-s + 25-s + 27-s + 8.74·29-s − 0.627·33-s + 3.37·35-s − 9.37·37-s − 39-s − 8.11·41-s + 10.7·43-s − 45-s + 4·47-s + 4.37·49-s − 5.37·51-s − 1.37·53-s + 0.627·55-s − 6.74·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.27·7-s + 0.333·9-s − 0.189·11-s − 0.277·13-s − 0.258·15-s − 1.30·17-s − 1.54·19-s − 0.735·21-s + 1.53·23-s + 0.200·25-s + 0.192·27-s + 1.62·29-s − 0.109·33-s + 0.570·35-s − 1.54·37-s − 0.160·39-s − 1.26·41-s + 1.63·43-s − 0.149·45-s + 0.583·47-s + 0.624·49-s − 0.752·51-s − 0.188·53-s + 0.0846·55-s − 0.893·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.301564845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.301564845\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 + 0.627T + 11T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 23 | \( 1 - 7.37T + 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9.37T + 37T^{2} \) |
| 41 | \( 1 + 8.11T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 1.37T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 2.62T + 61T^{2} \) |
| 67 | \( 1 + 9.48T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 3.37T + 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 + 9.37T + 89T^{2} \) |
| 97 | \( 1 - 5.37T + 97T^{2} \) |
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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
−8.187110035705169101670671685782, −7.16470395155086891787398953434, −6.75033611688615411215862387535, −6.19708465351487138075673695168, −4.97193819103969118621659914529, −4.37596372168201113290450963265, −3.50556840931358384532802721418, −2.84136018688262213149539336513, −2.08090317349652672795486402089, −0.54768202888856725153124558672,
0.54768202888856725153124558672, 2.08090317349652672795486402089, 2.84136018688262213149539336513, 3.50556840931358384532802721418, 4.37596372168201113290450963265, 4.97193819103969118621659914529, 6.19708465351487138075673695168, 6.75033611688615411215862387535, 7.16470395155086891787398953434, 8.187110035705169101670671685782
