| L(s) = 1 | + 3-s + 5-s − 2.80·7-s + 9-s − 5.68·11-s − 2.36·13-s + 15-s + 3.76·17-s + 4.21·19-s − 2.80·21-s + 0.258·23-s + 25-s + 27-s − 1.06·29-s + 4.94·31-s − 5.68·33-s − 2.80·35-s − 1.71·37-s − 2.36·39-s + 7.05·41-s + 1.19·43-s + 45-s + 12.8·47-s + 0.878·49-s + 3.76·51-s + 6.06·53-s − 5.68·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.06·7-s + 0.333·9-s − 1.71·11-s − 0.656·13-s + 0.258·15-s + 0.913·17-s + 0.967·19-s − 0.612·21-s + 0.0538·23-s + 0.200·25-s + 0.192·27-s − 0.197·29-s + 0.887·31-s − 0.989·33-s − 0.474·35-s − 0.281·37-s − 0.378·39-s + 1.10·41-s + 0.181·43-s + 0.149·45-s + 1.86·47-s + 0.125·49-s + 0.527·51-s + 0.833·53-s − 0.766·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.955314838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.955314838\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 2.80T + 7T^{2} \) |
| 11 | \( 1 + 5.68T + 11T^{2} \) |
| 13 | \( 1 + 2.36T + 13T^{2} \) |
| 17 | \( 1 - 3.76T + 17T^{2} \) |
| 19 | \( 1 - 4.21T + 19T^{2} \) |
| 23 | \( 1 - 0.258T + 23T^{2} \) |
| 29 | \( 1 + 1.06T + 29T^{2} \) |
| 31 | \( 1 - 4.94T + 31T^{2} \) |
| 37 | \( 1 + 1.71T + 37T^{2} \) |
| 41 | \( 1 - 7.05T + 41T^{2} \) |
| 43 | \( 1 - 1.19T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 - 6.06T + 53T^{2} \) |
| 59 | \( 1 + 5.14T + 59T^{2} \) |
| 61 | \( 1 - 0.0917T + 61T^{2} \) |
| 71 | \( 1 - 7.04T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 4.55T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 0.0460T + 89T^{2} \) |
| 97 | \( 1 - 8.86T + 97T^{2} \) |
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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
−8.420208197242174328110735807320, −7.56966893111208565857455668873, −7.27293712456642483176763875931, −6.14537321649189211234741560694, −5.49871039177349573580675478881, −4.78223421289476782622927754620, −3.59000046800083051986132785353, −2.84947636762531458057586867505, −2.31344606191308829215848706860, −0.75502098428600611087460402646,
0.75502098428600611087460402646, 2.31344606191308829215848706860, 2.84947636762531458057586867505, 3.59000046800083051986132785353, 4.78223421289476782622927754620, 5.49871039177349573580675478881, 6.14537321649189211234741560694, 7.27293712456642483176763875931, 7.56966893111208565857455668873, 8.420208197242174328110735807320
