| L(s) = 1 | − 2-s + 4-s + 5-s − 4.53·7-s − 8-s − 10-s − 6.53·11-s + 6·13-s + 4.53·14-s + 16-s + 4·17-s + 4.53·19-s + 20-s + 6.53·22-s − 6.53·23-s + 25-s − 6·26-s − 4.53·28-s + 31-s − 32-s − 4·34-s − 4.53·35-s − 7.06·37-s − 4.53·38-s − 40-s − 7.06·41-s − 8.53·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s − 1.71·7-s − 0.353·8-s − 0.316·10-s − 1.96·11-s + 1.66·13-s + 1.21·14-s + 0.250·16-s + 0.970·17-s + 1.03·19-s + 0.223·20-s + 1.39·22-s − 1.36·23-s + 0.200·25-s − 1.17·26-s − 0.856·28-s + 0.179·31-s − 0.176·32-s − 0.685·34-s − 0.765·35-s − 1.16·37-s − 0.735·38-s − 0.158·40-s − 1.10·41-s − 1.30·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8971697708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8971697708\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 4.53T + 7T^{2} \) |
| 11 | \( 1 + 6.53T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 4.53T + 19T^{2} \) |
| 23 | \( 1 + 6.53T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 37 | \( 1 + 7.06T + 37T^{2} \) |
| 41 | \( 1 + 7.06T + 41T^{2} \) |
| 43 | \( 1 + 8.53T + 43T^{2} \) |
| 47 | \( 1 - 5.06T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 - 9.06T + 59T^{2} \) |
| 61 | \( 1 - 5.06T + 61T^{2} \) |
| 67 | \( 1 - 5.06T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 + 8.53T + 73T^{2} \) |
| 79 | \( 1 + 8.53T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 - 6.53T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−8.748764625514085868370584798305, −8.184430508147482061749939701546, −7.35145870006563386466186940461, −6.54233965340957264836063288495, −5.80545879091466943363675361482, −5.31241979353152848698695725299, −3.55965533248184233470433661693, −3.17110080301766094079044645077, −2.05951540710162360912851985623, −0.62849609232271989998880696885,
0.62849609232271989998880696885, 2.05951540710162360912851985623, 3.17110080301766094079044645077, 3.55965533248184233470433661693, 5.31241979353152848698695725299, 5.80545879091466943363675361482, 6.54233965340957264836063288495, 7.35145870006563386466186940461, 8.184430508147482061749939701546, 8.748764625514085868370584798305
