| L(s) = 1 | + 2-s − 2.15·3-s + 4-s − 0.594·5-s − 2.15·6-s + 4.60·7-s + 8-s + 1.66·9-s − 0.594·10-s − 2.14·11-s − 2.15·12-s + 5.89·13-s + 4.60·14-s + 1.28·15-s + 16-s − 4.24·17-s + 1.66·18-s + 19-s − 0.594·20-s − 9.93·21-s − 2.14·22-s + 0.569·23-s − 2.15·24-s − 4.64·25-s + 5.89·26-s + 2.88·27-s + 4.60·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.24·3-s + 0.5·4-s − 0.265·5-s − 0.881·6-s + 1.73·7-s + 0.353·8-s + 0.554·9-s − 0.188·10-s − 0.645·11-s − 0.623·12-s + 1.63·13-s + 1.22·14-s + 0.331·15-s + 0.250·16-s − 1.03·17-s + 0.392·18-s + 0.229·19-s − 0.132·20-s − 2.16·21-s − 0.456·22-s + 0.118·23-s − 0.440·24-s − 0.929·25-s + 1.15·26-s + 0.555·27-s + 0.869·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.488155144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.488155144\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 2.15T + 3T^{2} \) |
| 5 | \( 1 + 0.594T + 5T^{2} \) |
| 7 | \( 1 - 4.60T + 7T^{2} \) |
| 11 | \( 1 + 2.14T + 11T^{2} \) |
| 13 | \( 1 - 5.89T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 23 | \( 1 - 0.569T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 - 3.56T + 37T^{2} \) |
| 41 | \( 1 - 1.57T + 41T^{2} \) |
| 43 | \( 1 + 0.914T + 43T^{2} \) |
| 47 | \( 1 - 7.85T + 47T^{2} \) |
| 53 | \( 1 - 8.68T + 53T^{2} \) |
| 59 | \( 1 - 9.36T + 59T^{2} \) |
| 61 | \( 1 - 6.01T + 61T^{2} \) |
| 67 | \( 1 + 3.84T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 5.64T + 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 + 1.01T + 83T^{2} \) |
| 89 | \( 1 - 7.50T + 89T^{2} \) |
| 97 | \( 1 + 4.05T + 97T^{2} \) |
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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
−7.74715580348720052548572057242, −7.02054714906073058161421752809, −6.15141583239911154241919999643, −5.64371728732231012432203183303, −5.16689951625329982687575974762, −4.31405401758693598286526226561, −3.98167702547093859060875413799, −2.62411626098953392775268899589, −1.69912413146259446455366300042, −0.78222718493580444551245549254,
0.78222718493580444551245549254, 1.69912413146259446455366300042, 2.62411626098953392775268899589, 3.98167702547093859060875413799, 4.31405401758693598286526226561, 5.16689951625329982687575974762, 5.64371728732231012432203183303, 6.15141583239911154241919999643, 7.02054714906073058161421752809, 7.74715580348720052548572057242
