| L(s) = 1 | + 4.87·3-s − 10.4·5-s + 23.3·7-s − 3.18·9-s + 23.8·11-s − 12.7·13-s − 51.1·15-s + 3.47·17-s + 33.0·19-s + 113.·21-s − 141.·23-s − 15.2·25-s − 147.·27-s + 29·29-s − 301.·31-s + 116.·33-s − 244.·35-s − 83.0·37-s − 62.3·39-s + 282.·41-s + 151.·43-s + 33.4·45-s − 138.·47-s + 200.·49-s + 16.9·51-s − 452.·53-s − 249.·55-s + ⋯ |
| L(s) = 1 | + 0.939·3-s − 0.937·5-s + 1.25·7-s − 0.118·9-s + 0.653·11-s − 0.272·13-s − 0.880·15-s + 0.0496·17-s + 0.399·19-s + 1.18·21-s − 1.28·23-s − 0.121·25-s − 1.05·27-s + 0.185·29-s − 1.74·31-s + 0.613·33-s − 1.17·35-s − 0.369·37-s − 0.256·39-s + 1.07·41-s + 0.537·43-s + 0.110·45-s − 0.430·47-s + 0.584·49-s + 0.0465·51-s − 1.17·53-s − 0.612·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 - 4.87T + 27T^{2} \) |
| 5 | \( 1 + 10.4T + 125T^{2} \) |
| 7 | \( 1 - 23.3T + 343T^{2} \) |
| 11 | \( 1 - 23.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 3.47T + 4.91e3T^{2} \) |
| 19 | \( 1 - 33.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 141.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 301.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 83.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 138.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 452.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 548.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 28.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 521.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 263.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 765.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 367.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 13.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 469.T + 9.12e5T^{2} \) |
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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
−8.411801432123869064417643021288, −7.73670823403082397080050613536, −7.39205562909249025920962277524, −6.05923177044546255635060875445, −5.09718409542321486383971041488, −4.09835144578778905967820082204, −3.58351772455205149355166883734, −2.37997804740801584917375860951, −1.48598136447226439093564673186, 0,
1.48598136447226439093564673186, 2.37997804740801584917375860951, 3.58351772455205149355166883734, 4.09835144578778905967820082204, 5.09718409542321486383971041488, 6.05923177044546255635060875445, 7.39205562909249025920962277524, 7.73670823403082397080050613536, 8.411801432123869064417643021288
