| L(s) = 1 | + 1.23·5-s + 4·7-s + 2.47·11-s + 4.47·13-s + 5.23·17-s − 3.23·23-s − 3.47·25-s − 3.70·29-s + 10.4·31-s + 4.94·35-s + 37-s − 6.94·41-s + 6.47·43-s − 8·47-s + 9·49-s + 0.472·53-s + 3.05·55-s − 12.1·59-s − 14.9·61-s + 5.52·65-s − 4.94·67-s − 12.9·71-s − 0.472·73-s + 9.88·77-s + 8.94·79-s + 5.52·83-s + 6.47·85-s + ⋯ |
| L(s) = 1 | + 0.552·5-s + 1.51·7-s + 0.745·11-s + 1.24·13-s + 1.26·17-s − 0.674·23-s − 0.694·25-s − 0.688·29-s + 1.88·31-s + 0.835·35-s + 0.164·37-s − 1.08·41-s + 0.986·43-s − 1.16·47-s + 1.28·49-s + 0.0648·53-s + 0.412·55-s − 1.58·59-s − 1.91·61-s + 0.685·65-s − 0.604·67-s − 1.53·71-s − 0.0552·73-s + 1.12·77-s + 1.00·79-s + 0.606·83-s + 0.702·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.845081396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.845081396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 + 4.94T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 0.472T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 5.52T + 83T^{2} \) |
| 89 | \( 1 + 4.29T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.782028571213032384292033192455, −8.044976487369645510405441228168, −7.59621024735117341019591045294, −6.28721564626949737831762944589, −5.90243686021373011354021344874, −4.92010125047906564554295498789, −4.15631154445076857531036946173, −3.18662125245793561674839628210, −1.79305340950460555158943807143, −1.24958232241972367064256834647,
1.24958232241972367064256834647, 1.79305340950460555158943807143, 3.18662125245793561674839628210, 4.15631154445076857531036946173, 4.92010125047906564554295498789, 5.90243686021373011354021344874, 6.28721564626949737831762944589, 7.59621024735117341019591045294, 8.044976487369645510405441228168, 8.782028571213032384292033192455
