| L(s) = 1 | + 9·3-s + 47.5·5-s − 19.5·7-s + 81·9-s − 562.·11-s − 169·13-s + 428.·15-s + 1.08e3·17-s + 1.22e3·19-s − 176.·21-s − 2.16e3·23-s − 860.·25-s + 729·27-s − 2.51e3·29-s + 3.68e3·31-s − 5.06e3·33-s − 931.·35-s + 1.57e4·37-s − 1.52e3·39-s + 1.59e4·41-s + 8.95e3·43-s + 3.85e3·45-s + 3.02e4·47-s − 1.64e4·49-s + 9.78e3·51-s + 2.21e4·53-s − 2.67e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.851·5-s − 0.151·7-s + 0.333·9-s − 1.40·11-s − 0.277·13-s + 0.491·15-s + 0.912·17-s + 0.781·19-s − 0.0872·21-s − 0.854·23-s − 0.275·25-s + 0.192·27-s − 0.556·29-s + 0.687·31-s − 0.809·33-s − 0.128·35-s + 1.89·37-s − 0.160·39-s + 1.48·41-s + 0.738·43-s + 0.283·45-s + 1.99·47-s − 0.977·49-s + 0.526·51-s + 1.08·53-s − 1.19·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.062330722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.062330722\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 13 | \( 1 + 169T \) |
| good | 5 | \( 1 - 47.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 19.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 562.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.08e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.22e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.16e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.51e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.57e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.59e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.95e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.02e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.21e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.02e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.95e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.61e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.81e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.18e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.88e4T + 8.58e9T^{2} \) |
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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
−9.864252781811406396656993083219, −9.134297168690218187016180482060, −7.87255080306656232349773507671, −7.52327407606657515501047539698, −6.02723164365227113170495039285, −5.44518057073700911663592432471, −4.19949997603866835152391238386, −2.88515947973490616583990901343, −2.21019134427601685390964369008, −0.815595732479606924417724901749,
0.815595732479606924417724901749, 2.21019134427601685390964369008, 2.88515947973490616583990901343, 4.19949997603866835152391238386, 5.44518057073700911663592432471, 6.02723164365227113170495039285, 7.52327407606657515501047539698, 7.87255080306656232349773507671, 9.134297168690218187016180482060, 9.864252781811406396656993083219
