| L(s) = 1 | + 0.624·3-s + 5-s − 7-s − 2.61·9-s + 11-s + 0.624·13-s + 0.624·15-s − 1.37·17-s − 2·19-s − 0.624·21-s − 6.36·23-s + 25-s − 3.50·27-s + 6·29-s − 3.73·31-s + 0.624·33-s − 35-s + 2.36·37-s + 0.389·39-s + 1.01·41-s + 7.11·43-s − 2.61·45-s + 1.37·47-s + 49-s − 0.858·51-s + 13.5·53-s + 55-s + ⋯ |
| L(s) = 1 | + 0.360·3-s + 0.447·5-s − 0.377·7-s − 0.870·9-s + 0.301·11-s + 0.173·13-s + 0.161·15-s − 0.333·17-s − 0.458·19-s − 0.136·21-s − 1.32·23-s + 0.200·25-s − 0.674·27-s + 1.11·29-s − 0.671·31-s + 0.108·33-s − 0.169·35-s + 0.388·37-s + 0.0624·39-s + 0.158·41-s + 1.08·43-s − 0.389·45-s + 0.200·47-s + 0.142·49-s − 0.120·51-s + 1.86·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.978411690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.978411690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 0.624T + 3T^{2} \) |
| 13 | \( 1 - 0.624T + 13T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6.36T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 3.73T + 31T^{2} \) |
| 37 | \( 1 - 2.36T + 37T^{2} \) |
| 41 | \( 1 - 1.01T + 41T^{2} \) |
| 43 | \( 1 - 7.11T + 43T^{2} \) |
| 47 | \( 1 - 1.37T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 - 14.9T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 7.61T + 67T^{2} \) |
| 71 | \( 1 + 6.49T + 71T^{2} \) |
| 73 | \( 1 - 3.87T + 73T^{2} \) |
| 79 | \( 1 + 2.36T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + 1.24T + 97T^{2} \) |
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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
−8.287349025481418631305476436180, −7.32440013663648093801352865958, −6.58398627129575327606374475334, −5.92062765724403974330303882340, −5.39337155329744205445317389465, −4.24987690755278443851433894911, −3.66529282191288235122743649373, −2.61336518388983319194772400459, −2.10571447311819171897656616514, −0.69998280018599509121954231977,
0.69998280018599509121954231977, 2.10571447311819171897656616514, 2.61336518388983319194772400459, 3.66529282191288235122743649373, 4.24987690755278443851433894911, 5.39337155329744205445317389465, 5.92062765724403974330303882340, 6.58398627129575327606374475334, 7.32440013663648093801352865958, 8.287349025481418631305476436180
