| L(s) = 1 | + 2-s − 0.414·3-s + 4-s + 3.41·5-s − 0.414·6-s + 8-s − 2.82·9-s + 3.41·10-s + 11-s − 0.414·12-s − 1.82·13-s − 1.41·15-s + 16-s + 7.65·17-s − 2.82·18-s + 3.41·19-s + 3.41·20-s + 22-s + 2.24·23-s − 0.414·24-s + 6.65·25-s − 1.82·26-s + 2.41·27-s − 8.65·29-s − 1.41·30-s + 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.239·3-s + 0.5·4-s + 1.52·5-s − 0.169·6-s + 0.353·8-s − 0.942·9-s + 1.07·10-s + 0.301·11-s − 0.119·12-s − 0.507·13-s − 0.365·15-s + 0.250·16-s + 1.85·17-s − 0.666·18-s + 0.783·19-s + 0.763·20-s + 0.213·22-s + 0.467·23-s − 0.0845·24-s + 1.33·25-s − 0.358·26-s + 0.464·27-s − 1.60·29-s − 0.258·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.969178156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.969178156\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 - 2.24T + 23T^{2} \) |
| 29 | \( 1 + 8.65T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 6.58T + 37T^{2} \) |
| 41 | \( 1 - 2.58T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 8.41T + 59T^{2} \) |
| 61 | \( 1 + 6.17T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 3.07T + 71T^{2} \) |
| 73 | \( 1 - 6.58T + 73T^{2} \) |
| 79 | \( 1 + 4.75T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 - 4.48T + 89T^{2} \) |
| 97 | \( 1 + 1.82T + 97T^{2} \) |
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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
−9.780669921061120277883462326499, −9.398488096627330637342785076954, −8.154331312975035835987980194374, −7.15887155261352747840362197028, −6.17740942356413052913480428200, −5.51398065353516102591224886369, −5.10235595335402172975087802136, −3.49481713111074960148953677764, −2.61889622135600741839840720979, −1.39756715963225636142486326199,
1.39756715963225636142486326199, 2.61889622135600741839840720979, 3.49481713111074960148953677764, 5.10235595335402172975087802136, 5.51398065353516102591224886369, 6.17740942356413052913480428200, 7.15887155261352747840362197028, 8.154331312975035835987980194374, 9.398488096627330637342785076954, 9.780669921061120277883462326499
