| L(s) = 1 | + 5.42·2-s + 1.67·3-s + 21.4·4-s + 7.70·5-s + 9.09·6-s − 15.0·7-s + 73.1·8-s − 24.1·9-s + 41.8·10-s − 2.51·11-s + 35.9·12-s − 81.5·14-s + 12.9·15-s + 225.·16-s − 2.06·17-s − 131.·18-s − 94.4·19-s + 165.·20-s − 25.1·21-s − 13.6·22-s + 35.8·23-s + 122.·24-s − 65.5·25-s − 85.7·27-s − 322.·28-s − 140.·29-s + 70.0·30-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 0.322·3-s + 2.68·4-s + 0.689·5-s + 0.618·6-s − 0.811·7-s + 3.23·8-s − 0.896·9-s + 1.32·10-s − 0.0688·11-s + 0.865·12-s − 1.55·14-s + 0.222·15-s + 3.51·16-s − 0.0294·17-s − 1.71·18-s − 1.14·19-s + 1.85·20-s − 0.261·21-s − 0.132·22-s + 0.325·23-s + 1.04·24-s − 0.524·25-s − 0.611·27-s − 2.17·28-s − 0.897·29-s + 0.426·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.655645941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.655645941\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 - 5.42T + 8T^{2} \) |
| 3 | \( 1 - 1.67T + 27T^{2} \) |
| 5 | \( 1 - 7.70T + 125T^{2} \) |
| 7 | \( 1 + 15.0T + 343T^{2} \) |
| 11 | \( 1 + 2.51T + 1.33e3T^{2} \) |
| 17 | \( 1 + 2.06T + 4.91e3T^{2} \) |
| 19 | \( 1 + 94.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 35.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 264.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 256.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 394.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 256.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 415.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 504.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 120.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 752.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 211.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 410.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 17.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 174.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 963.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 477.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.04e3T + 9.12e5T^{2} \) |
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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
−12.77038557953157083150371798028, −11.62305324453053864916425631538, −10.72079574405629055985577192505, −9.457959118218041427135719997246, −7.82622323704314033087963168265, −6.34593450644227549087318134209, −5.92783076292533781350701488233, −4.54140689987747881038407404565, −3.22174899349519804250096988547, −2.26268747917134585987484932300,
2.26268747917134585987484932300, 3.22174899349519804250096988547, 4.54140689987747881038407404565, 5.92783076292533781350701488233, 6.34593450644227549087318134209, 7.82622323704314033087963168265, 9.457959118218041427135719997246, 10.72079574405629055985577192505, 11.62305324453053864916425631538, 12.77038557953157083150371798028
