| L(s) = 1 | + 9.66·2-s − 1.55·3-s + 61.4·4-s + 25·5-s − 15.0·6-s + 220.·7-s + 285.·8-s − 240.·9-s + 241.·10-s − 121·11-s − 95.4·12-s + 913.·13-s + 2.13e3·14-s − 38.8·15-s + 788.·16-s + 1.89e3·17-s − 2.32e3·18-s − 361·19-s + 1.53e3·20-s − 342.·21-s − 1.16e3·22-s + 656.·23-s − 442.·24-s + 625·25-s + 8.83e3·26-s + 750.·27-s + 1.35e4·28-s + ⋯ |
| L(s) = 1 | + 1.70·2-s − 0.0995·3-s + 1.92·4-s + 0.447·5-s − 0.170·6-s + 1.70·7-s + 1.57·8-s − 0.990·9-s + 0.764·10-s − 0.301·11-s − 0.191·12-s + 1.49·13-s + 2.91·14-s − 0.0445·15-s + 0.770·16-s + 1.58·17-s − 1.69·18-s − 0.229·19-s + 0.859·20-s − 0.169·21-s − 0.515·22-s + 0.258·23-s − 0.156·24-s + 0.200·25-s + 2.56·26-s + 0.198·27-s + 3.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(9.635995403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.635995403\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 9.66T + 32T^{2} \) |
| 3 | \( 1 + 1.55T + 243T^{2} \) |
| 7 | \( 1 - 220.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 913.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.89e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 656.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.31e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.48e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.05e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.35e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.34e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.35e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.52e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.23e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.42e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.00e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.67e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.24e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.24e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.15e4T + 8.58e9T^{2} \) |
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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
−9.019633005281722563107081198406, −8.171553247925479229233157752727, −7.39650514413344371419927109744, −6.06944390429387401568957223849, −5.62983499480076225902839013930, −5.02422962846843952889448550417, −4.00633240641657749211297380431, −3.12668991543787224777972223918, −2.06736581718684831646293384365, −1.13563372715672122147370356217,
1.13563372715672122147370356217, 2.06736581718684831646293384365, 3.12668991543787224777972223918, 4.00633240641657749211297380431, 5.02422962846843952889448550417, 5.62983499480076225902839013930, 6.06944390429387401568957223849, 7.39650514413344371419927109744, 8.171553247925479229233157752727, 9.019633005281722563107081198406
