| L(s) = 1 | + 3.08·3-s − 0.786·5-s + 4.29·7-s + 6.51·9-s + 1.21·11-s − 5.08·13-s − 2.42·15-s − 2.29·17-s + 19-s + 13.2·21-s + 7.67·23-s − 4.38·25-s + 10.8·27-s + 0.489·29-s + 3.74·33-s − 3.38·35-s + 2·37-s − 15.6·39-s − 4.16·41-s − 12.9·43-s − 5.12·45-s − 5.80·47-s + 11.4·49-s − 7.08·51-s − 1.93·53-s − 0.954·55-s + 3.08·57-s + ⋯ |
| L(s) = 1 | + 1.78·3-s − 0.351·5-s + 1.62·7-s + 2.17·9-s + 0.365·11-s − 1.41·13-s − 0.626·15-s − 0.557·17-s + 0.229·19-s + 2.89·21-s + 1.60·23-s − 0.876·25-s + 2.08·27-s + 0.0909·29-s + 0.651·33-s − 0.571·35-s + 0.328·37-s − 2.51·39-s − 0.650·41-s − 1.97·43-s − 0.763·45-s − 0.847·47-s + 1.63·49-s − 0.991·51-s − 0.266·53-s − 0.128·55-s + 0.408·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.338800608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.338800608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 3.08T + 3T^{2} \) |
| 5 | \( 1 + 0.786T + 5T^{2} \) |
| 7 | \( 1 - 4.29T + 7T^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 13 | \( 1 + 5.08T + 13T^{2} \) |
| 17 | \( 1 + 2.29T + 17T^{2} \) |
| 23 | \( 1 - 7.67T + 23T^{2} \) |
| 29 | \( 1 - 0.489T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 4.16T + 41T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 47 | \( 1 + 5.80T + 47T^{2} \) |
| 53 | \( 1 + 1.93T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 5.38T + 61T^{2} \) |
| 67 | \( 1 + 2.48T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 8.46T + 73T^{2} \) |
| 79 | \( 1 - 1.83T + 79T^{2} \) |
| 83 | \( 1 - 7.02T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 3.57T + 97T^{2} \) |
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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
−9.530222174403217617299470884144, −8.732284547293606085633511129629, −8.191261480585018046320310898374, −7.49892888563301805436477418141, −6.91011933181459577598289719745, −5.05491666030481672902755041173, −4.51785979064268742069346790353, −3.45316413083897154696901349520, −2.41582926068248001397405527341, −1.56230628834506061117365327499,
1.56230628834506061117365327499, 2.41582926068248001397405527341, 3.45316413083897154696901349520, 4.51785979064268742069346790353, 5.05491666030481672902755041173, 6.91011933181459577598289719745, 7.49892888563301805436477418141, 8.191261480585018046320310898374, 8.732284547293606085633511129629, 9.530222174403217617299470884144
