| L(s) = 1 | + 24.5·3-s + 52.1·5-s + 141.·7-s + 357.·9-s − 53.4·11-s − 287.·13-s + 1.27e3·15-s + 338.·17-s − 290.·19-s + 3.47e3·21-s − 993.·23-s − 410.·25-s + 2.80e3·27-s − 5.50e3·29-s − 961·31-s − 1.30e3·33-s + 7.39e3·35-s + 9.67e3·37-s − 7.05e3·39-s + 1.88e4·41-s − 1.03e3·43-s + 1.86e4·45-s − 1.49e4·47-s + 3.31e3·49-s + 8.29e3·51-s + 1.58e4·53-s − 2.78e3·55-s + ⋯ |
| L(s) = 1 | + 1.57·3-s + 0.932·5-s + 1.09·7-s + 1.47·9-s − 0.133·11-s − 0.472·13-s + 1.46·15-s + 0.284·17-s − 0.184·19-s + 1.71·21-s − 0.391·23-s − 0.131·25-s + 0.740·27-s − 1.21·29-s − 0.179·31-s − 0.209·33-s + 1.01·35-s + 1.16·37-s − 0.742·39-s + 1.75·41-s − 0.0855·43-s + 1.37·45-s − 0.988·47-s + 0.196·49-s + 0.446·51-s + 0.773·53-s − 0.124·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.042329923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.042329923\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + 961T \) |
| good | 3 | \( 1 - 24.5T + 243T^{2} \) |
| 5 | \( 1 - 52.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 141.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 53.4T + 1.61e5T^{2} \) |
| 13 | \( 1 + 287.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 338.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 290.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 993.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.50e3T + 2.05e7T^{2} \) |
| 37 | \( 1 - 9.67e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.88e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.03e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.49e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.58e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.76e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.73e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.96e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.96e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.75e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.42e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.85e4T + 8.58e9T^{2} \) |
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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
−12.88470743992231167780497817049, −11.38664561997324812441704391217, −10.03877603178982887881250663384, −9.263014696517753682881328046206, −8.220537257254436137516222460548, −7.40938397788102005384390186806, −5.65110026217875176346876694008, −4.18171324569566739765782525161, −2.58554044321335663095507958696, −1.66444228502020080104937511009,
1.66444228502020080104937511009, 2.58554044321335663095507958696, 4.18171324569566739765782525161, 5.65110026217875176346876694008, 7.40938397788102005384390186806, 8.220537257254436137516222460548, 9.263014696517753682881328046206, 10.03877603178982887881250663384, 11.38664561997324812441704391217, 12.88470743992231167780497817049
