| L(s) = 1 | − 8.71·5-s + 7·7-s − 43.5·11-s + 82·13-s + 78.4·17-s + 20·19-s − 130.·23-s − 48.9·25-s − 244.·29-s − 156·31-s − 61.0·35-s + 186·37-s + 165.·41-s − 164·43-s + 470.·47-s + 49·49-s + 156.·53-s + 380.·55-s + 156.·59-s + 790·61-s − 714.·65-s + 44·67-s + 444.·71-s + 126·73-s − 305.·77-s + 712·79-s − 1.46e3·83-s + ⋯ |
| L(s) = 1 | − 0.779·5-s + 0.377·7-s − 1.19·11-s + 1.74·13-s + 1.11·17-s + 0.241·19-s − 1.18·23-s − 0.391·25-s − 1.56·29-s − 0.903·31-s − 0.294·35-s + 0.826·37-s + 0.630·41-s − 0.581·43-s + 1.46·47-s + 0.142·49-s + 0.406·53-s + 0.931·55-s + 0.346·59-s + 1.65·61-s − 1.36·65-s + 0.0802·67-s + 0.743·71-s + 0.202·73-s − 0.451·77-s + 1.01·79-s − 1.93·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.667971534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.667971534\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 8.71T + 125T^{2} \) |
| 11 | \( 1 + 43.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 82T + 2.19e3T^{2} \) |
| 17 | \( 1 - 78.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 20T + 6.85e3T^{2} \) |
| 23 | \( 1 + 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 244.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 156T + 2.97e4T^{2} \) |
| 37 | \( 1 - 186T + 5.06e4T^{2} \) |
| 41 | \( 1 - 165.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 164T + 7.95e4T^{2} \) |
| 47 | \( 1 - 470.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 156.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 156.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 790T + 2.26e5T^{2} \) |
| 67 | \( 1 - 44T + 3.00e5T^{2} \) |
| 71 | \( 1 - 444.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 126T + 3.89e5T^{2} \) |
| 79 | \( 1 - 712T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.46e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.45e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 798T + 9.12e5T^{2} \) |
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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
−9.611506322106140664925600804380, −8.530994636132838759177155973983, −7.898748749542417279039900327228, −7.37606399789814667655816734795, −5.93907304648391859968065387036, −5.42508272562905408319767873652, −4.04488388807169588258667382729, −3.48109701445526658662396672429, −2.04286806262321849012343336009, −0.68809536643117472969658959499,
0.68809536643117472969658959499, 2.04286806262321849012343336009, 3.48109701445526658662396672429, 4.04488388807169588258667382729, 5.42508272562905408319767873652, 5.93907304648391859968065387036, 7.37606399789814667655816734795, 7.898748749542417279039900327228, 8.530994636132838759177155973983, 9.611506322106140664925600804380
