| L(s) = 1 | + 4·2-s + 22.6·3-s + 16·4-s + 90.7·6-s + 49·7-s + 64·8-s + 272.·9-s + 527.·11-s + 363.·12-s + 1.08e3·13-s + 196·14-s + 256·16-s − 1.82e3·17-s + 1.08e3·18-s − 113.·19-s + 1.11e3·21-s + 2.11e3·22-s − 8.45·23-s + 1.45e3·24-s + 4.32e3·26-s + 660.·27-s + 784·28-s − 7.75e3·29-s − 5.07e3·31-s + 1.02e3·32-s + 1.19e4·33-s − 7.30e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.45·3-s + 0.5·4-s + 1.02·6-s + 0.377·7-s + 0.353·8-s + 1.11·9-s + 1.31·11-s + 0.727·12-s + 1.77·13-s + 0.267·14-s + 0.250·16-s − 1.53·17-s + 0.791·18-s − 0.0724·19-s + 0.550·21-s + 0.930·22-s − 0.00333·23-s + 0.514·24-s + 1.25·26-s + 0.174·27-s + 0.188·28-s − 1.71·29-s − 0.949·31-s + 0.176·32-s + 1.91·33-s − 1.08·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.625326950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.625326950\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 22.6T + 243T^{2} \) |
| 11 | \( 1 - 527.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.08e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.82e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 113.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 8.45T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.07e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.12e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.49e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.71e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.65e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.20e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 6.32e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.60e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.98e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.27e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.37e4T + 8.58e9T^{2} \) |
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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
−10.99643424090684682717226271445, −9.321409164105958891551479644556, −8.885518316416198260038046245709, −7.916382431603741694352613304359, −6.83295055284018971038360138388, −5.84329742944684401575417496558, −4.10293492810337277756771988167, −3.77441839996670738868391650268, −2.37365359515409725915515664714, −1.39375013771849860077858022766,
1.39375013771849860077858022766, 2.37365359515409725915515664714, 3.77441839996670738868391650268, 4.10293492810337277756771988167, 5.84329742944684401575417496558, 6.83295055284018971038360138388, 7.916382431603741694352613304359, 8.885518316416198260038046245709, 9.321409164105958891551479644556, 10.99643424090684682717226271445
