| L(s) = 1 | + 3.74·5-s + 5.29·11-s − 4.24·13-s + 3.74·17-s − 2.82·19-s − 5.29·23-s + 9·25-s + 5.29·29-s − 8.48·31-s + 4·37-s + 3.74·41-s − 8·43-s + 7.48·47-s + 10.5·53-s + 19.7·55-s + 7.48·59-s + 9.89·61-s − 15.8·65-s − 12·67-s + 15.8·71-s + 1.41·73-s + 4·79-s + 14.9·83-s + 14·85-s − 3.74·89-s − 10.5·95-s − 9.89·97-s + ⋯ |
| L(s) = 1 | + 1.67·5-s + 1.59·11-s − 1.17·13-s + 0.907·17-s − 0.648·19-s − 1.10·23-s + 1.80·25-s + 0.982·29-s − 1.52·31-s + 0.657·37-s + 0.584·41-s − 1.21·43-s + 1.09·47-s + 1.45·53-s + 2.66·55-s + 0.974·59-s + 1.26·61-s − 1.96·65-s − 1.46·67-s + 1.88·71-s + 0.165·73-s + 0.450·79-s + 1.64·83-s + 1.51·85-s − 0.396·89-s − 1.08·95-s − 1.00·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.168960491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.168960491\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 3.74T + 5T^{2} \) |
| 11 | \( 1 - 5.29T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 3.74T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 7.48T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 7.48T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 3.74T + 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−7.958330491534990688430172659820, −6.99227610664638323977485280415, −6.55966645689197574829536811282, −5.79071765551447663701664296131, −5.33706882625383848980618110027, −4.37465845947502123117845117777, −3.59566758787241583009063976354, −2.42985464080898893740576738446, −1.92585723524760274504635212207, −0.940960787235130843704839675487,
0.940960787235130843704839675487, 1.92585723524760274504635212207, 2.42985464080898893740576738446, 3.59566758787241583009063976354, 4.37465845947502123117845117777, 5.33706882625383848980618110027, 5.79071765551447663701664296131, 6.55966645689197574829536811282, 6.99227610664638323977485280415, 7.958330491534990688430172659820
