| L(s) = 1 | − 2.37·3-s − 3.87·5-s + 0.916·7-s + 2.63·9-s − 3.03·11-s − 3.70·13-s + 9.19·15-s − 0.890·17-s − 7.02·19-s − 2.17·21-s + 5.98·23-s + 10.0·25-s + 0.877·27-s + 3.54·29-s + 3.93·31-s + 7.21·33-s − 3.55·35-s + 4.15·37-s + 8.79·39-s + 0.368·41-s − 2.60·43-s − 10.1·45-s − 9.02·47-s − 6.15·49-s + 2.11·51-s + 1.51·53-s + 11.7·55-s + ⋯ |
| L(s) = 1 | − 1.36·3-s − 1.73·5-s + 0.346·7-s + 0.876·9-s − 0.916·11-s − 1.02·13-s + 2.37·15-s − 0.216·17-s − 1.61·19-s − 0.474·21-s + 1.24·23-s + 2.00·25-s + 0.168·27-s + 0.658·29-s + 0.707·31-s + 1.25·33-s − 0.600·35-s + 0.683·37-s + 1.40·39-s + 0.0575·41-s − 0.397·43-s − 1.51·45-s − 1.31·47-s − 0.879·49-s + 0.296·51-s + 0.208·53-s + 1.58·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 + 3.87T + 5T^{2} \) |
| 7 | \( 1 - 0.916T + 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 + 3.70T + 13T^{2} \) |
| 17 | \( 1 + 0.890T + 17T^{2} \) |
| 19 | \( 1 + 7.02T + 19T^{2} \) |
| 23 | \( 1 - 5.98T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 31 | \( 1 - 3.93T + 31T^{2} \) |
| 37 | \( 1 - 4.15T + 37T^{2} \) |
| 41 | \( 1 - 0.368T + 41T^{2} \) |
| 43 | \( 1 + 2.60T + 43T^{2} \) |
| 47 | \( 1 + 9.02T + 47T^{2} \) |
| 53 | \( 1 - 1.51T + 53T^{2} \) |
| 59 | \( 1 - 5.86T + 59T^{2} \) |
| 61 | \( 1 - 8.31T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 3.32T + 73T^{2} \) |
| 79 | \( 1 - 6.27T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 7.32T + 89T^{2} \) |
| 97 | \( 1 - 8.38T + 97T^{2} \) |
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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
−7.80976222625508396302233100243, −6.82670825260477437535674440887, −6.55560849642971053924901439345, −5.35605470903026633279772774006, −4.71668261178582858060734032960, −4.48241602719329452797937843572, −3.32626486209229401426175768383, −2.37726967365008169638990046215, −0.77003006207582272118793067215, 0,
0.77003006207582272118793067215, 2.37726967365008169638990046215, 3.32626486209229401426175768383, 4.48241602719329452797937843572, 4.71668261178582858060734032960, 5.35605470903026633279772774006, 6.55560849642971053924901439345, 6.82670825260477437535674440887, 7.80976222625508396302233100243
