| L(s) = 1 | − 2·5-s + 4.92·7-s + 2.15·11-s − 3.32·13-s − 5.91·17-s − 0.927·19-s + 7.63·23-s − 25-s + 1.59·29-s − 5.18·31-s − 9.84·35-s + 1.53·37-s + 9.71·41-s + 6.65·43-s − 5.72·47-s + 17.2·49-s + 9.24·53-s − 4.31·55-s − 5.78·59-s + 14.8·61-s + 6.65·65-s + 10.5·67-s − 5.51·71-s − 13.5·73-s + 10.6·77-s − 8.23·79-s − 7.97·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.85·7-s + 0.649·11-s − 0.922·13-s − 1.43·17-s − 0.212·19-s + 1.59·23-s − 0.200·25-s + 0.296·29-s − 0.930·31-s − 1.66·35-s + 0.251·37-s + 1.51·41-s + 1.01·43-s − 0.834·47-s + 2.45·49-s + 1.26·53-s − 0.581·55-s − 0.752·59-s + 1.90·61-s + 0.825·65-s + 1.29·67-s − 0.654·71-s − 1.58·73-s + 1.20·77-s − 0.926·79-s − 0.875·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.981237037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.981237037\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 + 3.32T + 13T^{2} \) |
| 17 | \( 1 + 5.91T + 17T^{2} \) |
| 19 | \( 1 + 0.927T + 19T^{2} \) |
| 23 | \( 1 - 7.63T + 23T^{2} \) |
| 29 | \( 1 - 1.59T + 29T^{2} \) |
| 31 | \( 1 + 5.18T + 31T^{2} \) |
| 37 | \( 1 - 1.53T + 37T^{2} \) |
| 41 | \( 1 - 9.71T + 41T^{2} \) |
| 43 | \( 1 - 6.65T + 43T^{2} \) |
| 47 | \( 1 + 5.72T + 47T^{2} \) |
| 53 | \( 1 - 9.24T + 53T^{2} \) |
| 59 | \( 1 + 5.78T + 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 5.51T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 8.23T + 79T^{2} \) |
| 83 | \( 1 + 7.97T + 83T^{2} \) |
| 89 | \( 1 - 8.41T + 89T^{2} \) |
| 97 | \( 1 - 5.34T + 97T^{2} \) |
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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
−8.043610436663286645378192510493, −7.32150259122230007069341973569, −7.02948998902466466320522221263, −5.84406734492128211482543189687, −4.97215536526650811547922277759, −4.45527393144613737258945312346, −3.94028393600128410470057689228, −2.64466689812934548514119343424, −1.86246907101348555792878078597, −0.75457279993641746987487924232,
0.75457279993641746987487924232, 1.86246907101348555792878078597, 2.64466689812934548514119343424, 3.94028393600128410470057689228, 4.45527393144613737258945312346, 4.97215536526650811547922277759, 5.84406734492128211482543189687, 7.02948998902466466320522221263, 7.32150259122230007069341973569, 8.043610436663286645378192510493
