| L(s) = 1 | + 3-s − 5-s + 0.670·7-s + 9-s + 3.78·11-s + 4.11·13-s − 15-s + 1.83·17-s + 5.73·19-s + 0.670·21-s + 7.13·23-s + 25-s + 27-s − 1.82·29-s − 5.12·31-s + 3.78·33-s − 0.670·35-s + 2.09·37-s + 4.11·39-s + 7.12·41-s − 3.02·43-s − 45-s − 5.47·47-s − 6.55·49-s + 1.83·51-s + 6.86·53-s − 3.78·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.253·7-s + 0.333·9-s + 1.14·11-s + 1.14·13-s − 0.258·15-s + 0.446·17-s + 1.31·19-s + 0.146·21-s + 1.48·23-s + 0.200·25-s + 0.192·27-s − 0.338·29-s − 0.919·31-s + 0.658·33-s − 0.113·35-s + 0.343·37-s + 0.658·39-s + 1.11·41-s − 0.461·43-s − 0.149·45-s − 0.797·47-s − 0.935·49-s + 0.257·51-s + 0.942·53-s − 0.510·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.264714345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.264714345\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.670T + 7T^{2} \) |
| 11 | \( 1 - 3.78T + 11T^{2} \) |
| 13 | \( 1 - 4.11T + 13T^{2} \) |
| 17 | \( 1 - 1.83T + 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 23 | \( 1 - 7.13T + 23T^{2} \) |
| 29 | \( 1 + 1.82T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 - 2.09T + 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 + 5.47T + 47T^{2} \) |
| 53 | \( 1 - 6.86T + 53T^{2} \) |
| 59 | \( 1 - 1.63T + 59T^{2} \) |
| 61 | \( 1 - 5.50T + 61T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 7.01T + 79T^{2} \) |
| 83 | \( 1 + 5.15T + 83T^{2} \) |
| 89 | \( 1 - 5.63T + 89T^{2} \) |
| 97 | \( 1 + 9.80T + 97T^{2} \) |
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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
−7.78397351098181048072668861713, −7.25528103838298819277780243592, −6.58870059276422346169786597266, −5.74208406811642236235587336991, −4.98861992099895170164888636551, −4.07573056471976987794517691257, −3.51913414037541566306430749201, −2.89761550669903835452931355758, −1.57131317624822094289778061884, −0.983596852258812971532847292287,
0.983596852258812971532847292287, 1.57131317624822094289778061884, 2.89761550669903835452931355758, 3.51913414037541566306430749201, 4.07573056471976987794517691257, 4.98861992099895170164888636551, 5.74208406811642236235587336991, 6.58870059276422346169786597266, 7.25528103838298819277780243592, 7.78397351098181048072668861713
