L(s) = 1 | − 3-s − 4.27·5-s + 9-s + 4.27·11-s − 1.27·13-s + 4.27·15-s + 4·17-s + 1.27·19-s + 4·23-s + 13.2·25-s − 27-s + 2.27·29-s + 31-s − 4.27·33-s − 5.27·37-s + 1.27·39-s − 10.5·41-s + 7.27·43-s − 4.27·45-s + 6·47-s − 4·51-s − 1.72·53-s − 18.2·55-s − 1.27·57-s + 6.27·59-s + 10·61-s + 5.45·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.91·5-s + 0.333·9-s + 1.28·11-s − 0.353·13-s + 1.10·15-s + 0.970·17-s + 0.292·19-s + 0.834·23-s + 2.65·25-s − 0.192·27-s + 0.422·29-s + 0.179·31-s − 0.744·33-s − 0.867·37-s + 0.204·39-s − 1.64·41-s + 1.10·43-s − 0.637·45-s + 0.875·47-s − 0.560·51-s − 0.236·53-s − 2.46·55-s − 0.168·57-s + 0.816·59-s + 1.28·61-s + 0.676·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.139636305\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.139636305\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 4.27T + 5T^{2} \) |
| 11 | \( 1 - 4.27T + 11T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2.27T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 5.27T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 7.27T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 1.72T + 53T^{2} \) |
| 59 | \( 1 - 6.27T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 7.27T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 3.27T + 73T^{2} \) |
| 79 | \( 1 + 3.54T + 79T^{2} \) |
| 83 | \( 1 + 0.274T + 83T^{2} \) |
| 89 | \( 1 + 4.54T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−7.52163005416145884995657889863, −7.08523174645885749201228108695, −6.58454568030360544510996201111, −5.53246300652978250830600078117, −4.86138220182874694601741093827, −4.13032497675797827736617601388, −3.62631428653427476413652113246, −2.89785362068929057017137540454, −1.35309396468921934033987633481, −0.59221846577716553688416621360,
0.59221846577716553688416621360, 1.35309396468921934033987633481, 2.89785362068929057017137540454, 3.62631428653427476413652113246, 4.13032497675797827736617601388, 4.86138220182874694601741093827, 5.53246300652978250830600078117, 6.58454568030360544510996201111, 7.08523174645885749201228108695, 7.52163005416145884995657889863