| L(s) = 1 | + 10.5·5-s − 8·7-s − 15.8·11-s − 52.9·13-s + 14·17-s − 37.0·19-s + 152·23-s − 12.9·25-s − 158.·29-s − 224·31-s − 84.6·35-s + 243.·37-s − 70·41-s + 439.·43-s + 336·47-s − 279·49-s − 31.7·53-s − 168.·55-s + 534.·59-s − 95.2·61-s − 560·65-s + 174.·67-s + 72·71-s + 294·73-s + 126.·77-s + 464·79-s + 545.·83-s + ⋯ |
| L(s) = 1 | + 0.946·5-s − 0.431·7-s − 0.435·11-s − 1.12·13-s + 0.199·17-s − 0.447·19-s + 1.37·23-s − 0.103·25-s − 1.01·29-s − 1.29·31-s − 0.408·35-s + 1.08·37-s − 0.266·41-s + 1.55·43-s + 1.04·47-s − 0.813·49-s − 0.0822·53-s − 0.411·55-s + 1.17·59-s − 0.199·61-s − 1.06·65-s + 0.318·67-s + 0.120·71-s + 0.471·73-s + 0.187·77-s + 0.660·79-s + 0.720·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.020526253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.020526253\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 10.5T + 125T^{2} \) |
| 7 | \( 1 + 8T + 343T^{2} \) |
| 11 | \( 1 + 15.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 14T + 4.91e3T^{2} \) |
| 19 | \( 1 + 37.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 152T + 1.21e4T^{2} \) |
| 29 | \( 1 + 158.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 224T + 2.97e4T^{2} \) |
| 37 | \( 1 - 243.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 70T + 6.89e4T^{2} \) |
| 43 | \( 1 - 439.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 336T + 1.03e5T^{2} \) |
| 53 | \( 1 + 31.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 534.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 95.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 174.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 72T + 3.57e5T^{2} \) |
| 73 | \( 1 - 294T + 3.89e5T^{2} \) |
| 79 | \( 1 - 464T + 4.93e5T^{2} \) |
| 83 | \( 1 - 545.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 266T + 7.04e5T^{2} \) |
| 97 | \( 1 - 994T + 9.12e5T^{2} \) |
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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
−8.914259480843708520911077953070, −7.69683939139059565028775220730, −7.19007551491266636916664528955, −6.22643550336349782975399358224, −5.52752619019858416171652727477, −4.84830096115872763715833244027, −3.70872107119746542557986346240, −2.64031488528006242987927580777, −1.98113584395514724863790996878, −0.61368247359817023210203473703,
0.61368247359817023210203473703, 1.98113584395514724863790996878, 2.64031488528006242987927580777, 3.70872107119746542557986346240, 4.84830096115872763715833244027, 5.52752619019858416171652727477, 6.22643550336349782975399358224, 7.19007551491266636916664528955, 7.69683939139059565028775220730, 8.914259480843708520911077953070
