L(s) = 1 | + 5-s + 7-s + 4.44·11-s + 2.44·13-s + 3.44·17-s + 7.34·19-s − 23-s + 25-s + 9.44·29-s − 1.89·31-s + 35-s − 9.89·37-s + 0.550·41-s − 7.79·43-s − 7.34·47-s − 6·49-s − 4.34·53-s + 4.44·55-s + 6.55·59-s + 0.449·61-s + 2.44·65-s + 8.79·67-s − 2.34·71-s − 7.34·73-s + 4.44·77-s − 13.7·79-s + 15.4·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.34·11-s + 0.679·13-s + 0.836·17-s + 1.68·19-s − 0.208·23-s + 0.200·25-s + 1.75·29-s − 0.341·31-s + 0.169·35-s − 1.62·37-s + 0.0859·41-s − 1.18·43-s − 1.07·47-s − 0.857·49-s − 0.597·53-s + 0.599·55-s + 0.852·59-s + 0.0575·61-s + 0.303·65-s + 1.07·67-s − 0.278·71-s − 0.860·73-s + 0.507·77-s − 1.55·79-s + 1.69·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.792416112\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.792416112\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 3.44T + 17T^{2} \) |
| 19 | \( 1 - 7.34T + 19T^{2} \) |
| 29 | \( 1 - 9.44T + 29T^{2} \) |
| 31 | \( 1 + 1.89T + 31T^{2} \) |
| 37 | \( 1 + 9.89T + 37T^{2} \) |
| 41 | \( 1 - 0.550T + 41T^{2} \) |
| 43 | \( 1 + 7.79T + 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 + 4.34T + 53T^{2} \) |
| 59 | \( 1 - 6.55T + 59T^{2} \) |
| 61 | \( 1 - 0.449T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 + 2.34T + 71T^{2} \) |
| 73 | \( 1 + 7.34T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 3.10T + 89T^{2} \) |
| 97 | \( 1 - 4.89T + 97T^{2} \) |
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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
−8.486051677170449175993952635482, −7.73306771042633936488369791715, −6.82108938464682541674503827495, −6.31615433910063084842179603795, −5.39450616536368769383489325358, −4.80136844867913926214685672314, −3.66026892071206253258113392441, −3.13282321998270431888605023045, −1.69906004859066753445086354474, −1.08233632397486074791530573349,
1.08233632397486074791530573349, 1.69906004859066753445086354474, 3.13282321998270431888605023045, 3.66026892071206253258113392441, 4.80136844867913926214685672314, 5.39450616536368769383489325358, 6.31615433910063084842179603795, 6.82108938464682541674503827495, 7.73306771042633936488369791715, 8.486051677170449175993952635482