| L(s) = 1 | + 0.801·5-s − 3.69·7-s − 2.85·11-s − 4.44·17-s − 2.26·19-s − 7.78·23-s − 4.35·25-s − 0.246·29-s + 0.466·31-s − 2.96·35-s + 6.65·37-s + 8.63·41-s − 9.76·43-s + 11.8·47-s + 6.63·49-s + 8.94·53-s − 2.28·55-s + 0.396·59-s + 10.5·61-s − 3.30·67-s + 13.3·71-s + 10.2·73-s + 10.5·77-s − 9.72·79-s + 12.8·83-s − 3.56·85-s − 7.77·89-s + ⋯ |
| L(s) = 1 | + 0.358·5-s − 1.39·7-s − 0.859·11-s − 1.07·17-s − 0.520·19-s − 1.62·23-s − 0.871·25-s − 0.0458·29-s + 0.0838·31-s − 0.500·35-s + 1.09·37-s + 1.34·41-s − 1.48·43-s + 1.73·47-s + 0.947·49-s + 1.22·53-s − 0.308·55-s + 0.0515·59-s + 1.35·61-s − 0.404·67-s + 1.58·71-s + 1.20·73-s + 1.19·77-s − 1.09·79-s + 1.40·83-s − 0.386·85-s − 0.824·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9434769416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9434769416\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 0.801T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 17 | \( 1 + 4.44T + 17T^{2} \) |
| 19 | \( 1 + 2.26T + 19T^{2} \) |
| 23 | \( 1 + 7.78T + 23T^{2} \) |
| 29 | \( 1 + 0.246T + 29T^{2} \) |
| 31 | \( 1 - 0.466T + 31T^{2} \) |
| 37 | \( 1 - 6.65T + 37T^{2} \) |
| 41 | \( 1 - 8.63T + 41T^{2} \) |
| 43 | \( 1 + 9.76T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 8.94T + 53T^{2} \) |
| 59 | \( 1 - 0.396T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 3.30T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 9.72T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 7.77T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.091236923985679933799512376719, −7.31325314353737224626557176154, −6.51119944268772193648052930640, −6.04113888873097358528754497791, −5.38685824639437914103315778439, −4.25804955851733386007930921873, −3.73220700853308205532487354201, −2.57161478417333546831835178102, −2.16622941904472062861526105042, −0.47332887970073024724639746571,
0.47332887970073024724639746571, 2.16622941904472062861526105042, 2.57161478417333546831835178102, 3.73220700853308205532487354201, 4.25804955851733386007930921873, 5.38685824639437914103315778439, 6.04113888873097358528754497791, 6.51119944268772193648052930640, 7.31325314353737224626557176154, 8.091236923985679933799512376719
