| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 5.34·5-s − 36·6-s − 243.·7-s + 64·8-s + 81·9-s − 21.3·10-s − 601.·11-s − 144·12-s + 471.·13-s − 974.·14-s + 48.1·15-s + 256·16-s − 637.·17-s + 324·18-s − 14.4·19-s − 85.5·20-s + 2.19e3·21-s − 2.40e3·22-s + 3.72e3·23-s − 576·24-s − 3.09e3·25-s + 1.88e3·26-s − 729·27-s − 3.89e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0956·5-s − 0.408·6-s − 1.87·7-s + 0.353·8-s + 0.333·9-s − 0.0676·10-s − 1.49·11-s − 0.288·12-s + 0.773·13-s − 1.32·14-s + 0.0552·15-s + 0.250·16-s − 0.534·17-s + 0.235·18-s − 0.00917·19-s − 0.0478·20-s + 1.08·21-s − 1.06·22-s + 1.46·23-s − 0.204·24-s − 0.990·25-s + 0.546·26-s − 0.192·27-s − 0.939·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.535392997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.535392997\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 + 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 5 | \( 1 + 5.34T + 3.12e3T^{2} \) |
| 7 | \( 1 + 243.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 601.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 471.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 637.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 14.4T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.72e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.37e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.39e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.70e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.65e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.57e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 883.T + 4.18e8T^{2} \) |
| 61 | \( 1 - 2.03e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.30e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.92e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.97e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.02e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.13e5T + 8.58e9T^{2} \) |
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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
−10.61991277256055799770130883644, −10.05944080415910532352361536779, −8.855848666135384732457511093004, −7.48208097555680556983501096695, −6.52109331742521528861393322752, −5.87286322960400329471492632297, −4.76280693607853011829336272642, −3.48126065577312521003868710234, −2.59908812513642788703160651915, −0.59518686910823860836127115199,
0.59518686910823860836127115199, 2.59908812513642788703160651915, 3.48126065577312521003868710234, 4.76280693607853011829336272642, 5.87286322960400329471492632297, 6.52109331742521528861393322752, 7.48208097555680556983501096695, 8.855848666135384732457511093004, 10.05944080415910532352361536779, 10.61991277256055799770130883644
