| L(s) = 1 | + 5.85·3-s + 10.3·5-s + 30.9·7-s + 7.31·9-s + 19.6·11-s − 5.88·13-s + 60.8·15-s − 4.34·19-s + 181.·21-s + 71.8·23-s − 17.2·25-s − 115.·27-s + 149.·29-s − 54.3·31-s + 115.·33-s + 321.·35-s + 176.·37-s − 34.4·39-s − 337.·41-s + 539.·43-s + 75.9·45-s − 122.·47-s + 614.·49-s − 560.·53-s + 204.·55-s − 25.4·57-s + 711.·59-s + ⋯ |
| L(s) = 1 | + 1.12·3-s + 0.928·5-s + 1.67·7-s + 0.271·9-s + 0.539·11-s − 0.125·13-s + 1.04·15-s − 0.0524·19-s + 1.88·21-s + 0.651·23-s − 0.137·25-s − 0.821·27-s + 0.955·29-s − 0.314·31-s + 0.608·33-s + 1.55·35-s + 0.782·37-s − 0.141·39-s − 1.28·41-s + 1.91·43-s + 0.251·45-s − 0.380·47-s + 1.79·49-s − 1.45·53-s + 0.500·55-s − 0.0590·57-s + 1.57·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.679023456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.679023456\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 5.85T + 27T^{2} \) |
| 5 | \( 1 - 10.3T + 125T^{2} \) |
| 7 | \( 1 - 30.9T + 343T^{2} \) |
| 11 | \( 1 - 19.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5.88T + 2.19e3T^{2} \) |
| 19 | \( 1 + 4.34T + 6.85e3T^{2} \) |
| 23 | \( 1 - 71.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 54.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 176.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 539.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 122.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 560.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 711.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 439.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 519.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 863.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 589.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.28e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.14e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.13e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.61e3T + 9.12e5T^{2} \) |
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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
−8.620785146530149133219139266592, −8.055434934007817236138776416814, −7.34355944412238085364747439547, −6.31739525514924479503921331176, −5.39700728871644352126814082922, −4.66720946926767566608652823214, −3.71456119266698824971079300107, −2.57287534707790538868113717021, −1.94915270506660320440963676340, −1.09450600114803223972364635033,
1.09450600114803223972364635033, 1.94915270506660320440963676340, 2.57287534707790538868113717021, 3.71456119266698824971079300107, 4.66720946926767566608652823214, 5.39700728871644352126814082922, 6.31739525514924479503921331176, 7.34355944412238085364747439547, 8.055434934007817236138776416814, 8.620785146530149133219139266592
