| L(s) = 1 | − 2.46·3-s − 4.03·5-s + 3.10·9-s + 1.60·11-s + 3.15·13-s + 9.97·15-s + 0.652·17-s + 19-s + 2.91·23-s + 11.3·25-s − 0.248·27-s + 2.12·29-s − 7.02·31-s − 3.95·33-s − 5.30·37-s − 7.79·39-s − 2·41-s + 7.97·43-s − 12.5·45-s − 0.605·47-s − 1.61·51-s + 12.4·53-s − 6.47·55-s − 2.46·57-s + 14.4·59-s + 9.34·61-s − 12.7·65-s + ⋯ |
| L(s) = 1 | − 1.42·3-s − 1.80·5-s + 1.03·9-s + 0.483·11-s + 0.874·13-s + 2.57·15-s + 0.158·17-s + 0.229·19-s + 0.608·23-s + 2.26·25-s − 0.0477·27-s + 0.393·29-s − 1.26·31-s − 0.688·33-s − 0.872·37-s − 1.24·39-s − 0.312·41-s + 1.21·43-s − 1.86·45-s − 0.0883·47-s − 0.225·51-s + 1.71·53-s − 0.872·55-s − 0.327·57-s + 1.88·59-s + 1.19·61-s − 1.58·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6718498079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6718498079\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.46T + 3T^{2} \) |
| 5 | \( 1 + 4.03T + 5T^{2} \) |
| 11 | \( 1 - 1.60T + 11T^{2} \) |
| 13 | \( 1 - 3.15T + 13T^{2} \) |
| 17 | \( 1 - 0.652T + 17T^{2} \) |
| 23 | \( 1 - 2.91T + 23T^{2} \) |
| 29 | \( 1 - 2.12T + 29T^{2} \) |
| 31 | \( 1 + 7.02T + 31T^{2} \) |
| 37 | \( 1 + 5.30T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 7.97T + 43T^{2} \) |
| 47 | \( 1 + 0.605T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 9.34T + 61T^{2} \) |
| 67 | \( 1 + 2.78T + 67T^{2} \) |
| 71 | \( 1 + 9.50T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 + 8.73T + 83T^{2} \) |
| 89 | \( 1 + 2.27T + 89T^{2} \) |
| 97 | \( 1 - 1.24T + 97T^{2} \) |
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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
−7.71956834490964933757300386331, −7.04072708913272417851923800090, −6.68836722709374113914995997174, −5.64930909257193306321698241465, −5.21136181181800599848132520073, −4.16688407413964735900730539683, −3.90862994319465995814340749311, −2.93966823292409946157197193714, −1.29095645664281311727707889651, −0.50755109545045332146904247087,
0.50755109545045332146904247087, 1.29095645664281311727707889651, 2.93966823292409946157197193714, 3.90862994319465995814340749311, 4.16688407413964735900730539683, 5.21136181181800599848132520073, 5.64930909257193306321698241465, 6.68836722709374113914995997174, 7.04072708913272417851923800090, 7.71956834490964933757300386331
