| L(s) = 1 | + 2.56·5-s + 3.12·7-s + 6.56·11-s + 0.561·13-s + 17-s − 7.68·19-s + 3.43·23-s + 1.56·25-s − 1.12·29-s − 8.24·31-s + 8·35-s − 4·37-s − 9.68·41-s + 7.68·43-s + 9.12·47-s + 2.75·49-s − 6·53-s + 16.8·55-s − 11.3·59-s − 4·61-s + 1.43·65-s + 12·67-s + 13.3·71-s − 8.24·73-s + 20.4·77-s + 2·79-s + 1.12·83-s + ⋯ |
| L(s) = 1 | + 1.14·5-s + 1.18·7-s + 1.97·11-s + 0.155·13-s + 0.242·17-s − 1.76·19-s + 0.716·23-s + 0.312·25-s − 0.208·29-s − 1.48·31-s + 1.35·35-s − 0.657·37-s − 1.51·41-s + 1.17·43-s + 1.33·47-s + 0.393·49-s − 0.824·53-s + 2.26·55-s − 1.48·59-s − 0.512·61-s + 0.178·65-s + 1.46·67-s + 1.58·71-s − 0.965·73-s + 2.33·77-s + 0.225·79-s + 0.123·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.499479078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.499479078\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 - 3.12T + 7T^{2} \) |
| 11 | \( 1 - 6.56T + 11T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 19 | \( 1 + 7.68T + 19T^{2} \) |
| 23 | \( 1 - 3.43T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 31 | \( 1 + 8.24T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 9.68T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 - 9.12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 - 1.12T + 83T^{2} \) |
| 89 | \( 1 + 0.876T + 89T^{2} \) |
| 97 | \( 1 + 7.12T + 97T^{2} \) |
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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
−9.505824341834268588500396389776, −9.022029750026767749609063501177, −8.294182296127050839801587132244, −7.09585878267292586787741440806, −6.36649664227405469273324000501, −5.58099321793151336457307091847, −4.58404681245385885261275425394, −3.69456300146084032782259820902, −2.05281782700434321431250394155, −1.43677037652762950295931386227,
1.43677037652762950295931386227, 2.05281782700434321431250394155, 3.69456300146084032782259820902, 4.58404681245385885261275425394, 5.58099321793151336457307091847, 6.36649664227405469273324000501, 7.09585878267292586787741440806, 8.294182296127050839801587132244, 9.022029750026767749609063501177, 9.505824341834268588500396389776
