| L(s) = 1 | + 1.34·5-s − 7-s − 6.28·11-s − 4.43·13-s + 0.505·17-s + 6.19·19-s + 5.84·23-s − 3.19·25-s + 4.43·29-s + 6.62·31-s − 1.34·35-s − 8.44·37-s − 5.77·41-s − 7.44·43-s + 12.1·47-s + 49-s − 13.8·53-s − 8.44·55-s + 6.25·59-s + 1.01·61-s − 5.95·65-s + 14.8·67-s − 2.35·71-s + 11.8·73-s + 6.28·77-s − 12.3·79-s + 8.43·83-s + ⋯ |
| L(s) = 1 | + 0.601·5-s − 0.377·7-s − 1.89·11-s − 1.22·13-s + 0.122·17-s + 1.42·19-s + 1.21·23-s − 0.638·25-s + 0.822·29-s + 1.18·31-s − 0.227·35-s − 1.38·37-s − 0.901·41-s − 1.13·43-s + 1.76·47-s + 0.142·49-s − 1.90·53-s − 1.13·55-s + 0.814·59-s + 0.129·61-s − 0.738·65-s + 1.81·67-s − 0.279·71-s + 1.38·73-s + 0.715·77-s − 1.38·79-s + 0.925·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.555702562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.555702562\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 1.34T + 5T^{2} \) |
| 11 | \( 1 + 6.28T + 11T^{2} \) |
| 13 | \( 1 + 4.43T + 13T^{2} \) |
| 17 | \( 1 - 0.505T + 17T^{2} \) |
| 19 | \( 1 - 6.19T + 19T^{2} \) |
| 23 | \( 1 - 5.84T + 23T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 31 | \( 1 - 6.62T + 31T^{2} \) |
| 37 | \( 1 + 8.44T + 37T^{2} \) |
| 41 | \( 1 + 5.77T + 41T^{2} \) |
| 43 | \( 1 + 7.44T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 6.25T + 59T^{2} \) |
| 61 | \( 1 - 1.01T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 2.35T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 8.43T + 83T^{2} \) |
| 89 | \( 1 + 4.91T + 89T^{2} \) |
| 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
−8.016579163309820063836838268095, −7.33263484829767509449256210010, −6.79540163254483441722402074487, −5.78574170077626221599799621330, −5.08998318562229181139493506850, −4.85812261577029917335405693185, −3.33834685511071289423460235521, −2.80554561691882405838896693625, −2.02236334180168580352117132280, −0.62809947657718713706820553989,
0.62809947657718713706820553989, 2.02236334180168580352117132280, 2.80554561691882405838896693625, 3.33834685511071289423460235521, 4.85812261577029917335405693185, 5.08998318562229181139493506850, 5.78574170077626221599799621330, 6.79540163254483441722402074487, 7.33263484829767509449256210010, 8.016579163309820063836838268095
