| L(s) = 1 | + 3.37·2-s − 3·3-s + 3.40·4-s − 15.7·5-s − 10.1·6-s − 17.1·7-s − 15.5·8-s + 9·9-s − 53.0·10-s + 52.8·11-s − 10.2·12-s − 57.9·14-s + 47.1·15-s − 79.6·16-s − 71.0·17-s + 30.3·18-s + 92.6·19-s − 53.5·20-s + 51.5·21-s + 178.·22-s + 190.·23-s + 46.5·24-s + 121.·25-s − 27·27-s − 58.4·28-s − 128.·29-s + 159.·30-s + ⋯ |
| L(s) = 1 | + 1.19·2-s − 0.577·3-s + 0.425·4-s − 1.40·5-s − 0.689·6-s − 0.926·7-s − 0.685·8-s + 0.333·9-s − 1.67·10-s + 1.44·11-s − 0.245·12-s − 1.10·14-s + 0.811·15-s − 1.24·16-s − 1.01·17-s + 0.398·18-s + 1.11·19-s − 0.598·20-s + 0.535·21-s + 1.72·22-s + 1.72·23-s + 0.395·24-s + 0.975·25-s − 0.192·27-s − 0.394·28-s − 0.820·29-s + 0.968·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.631552260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.631552260\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 3.37T + 8T^{2} \) |
| 5 | \( 1 + 15.7T + 125T^{2} \) |
| 7 | \( 1 + 17.1T + 343T^{2} \) |
| 11 | \( 1 - 52.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 71.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 92.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 190.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 3.29T + 2.97e4T^{2} \) |
| 37 | \( 1 - 241.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 97.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 376.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 577.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 307.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 349.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 127.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 903.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 826.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 131.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 556.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 254.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 183.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 780.T + 9.12e5T^{2} \) |
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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
−11.11050661898144754400286151316, −9.467643882067808371855349475395, −8.897906987540019031362538093204, −7.32876204862082960893900153785, −6.70080922344222622883381992866, −5.74243906758899192447704400528, −4.54090400045835461265662704072, −3.90288222481510906834762943886, −3.04957979983645306198235924513, −0.67410225259641778215507783515,
0.67410225259641778215507783515, 3.04957979983645306198235924513, 3.90288222481510906834762943886, 4.54090400045835461265662704072, 5.74243906758899192447704400528, 6.70080922344222622883381992866, 7.32876204862082960893900153785, 8.897906987540019031362538093204, 9.467643882067808371855349475395, 11.11050661898144754400286151316
