| L(s) = 1 | − 20.8·3-s + 11.6·5-s + 191.·9-s − 556.·11-s − 122.·13-s − 242.·15-s + 756.·17-s − 1.23e3·19-s + 1.44e3·23-s − 2.98e3·25-s + 1.06e3·27-s + 1.01e3·29-s − 1.01e4·31-s + 1.15e4·33-s + 6.49e3·37-s + 2.54e3·39-s − 1.25e4·41-s − 1.38e3·43-s + 2.23e3·45-s − 1.16e4·47-s − 1.57e4·51-s + 1.74e4·53-s − 6.47e3·55-s + 2.58e4·57-s − 3.40e4·59-s + 2.77e4·61-s − 1.42e3·65-s + ⋯ |
| L(s) = 1 | − 1.33·3-s + 0.208·5-s + 0.788·9-s − 1.38·11-s − 0.200·13-s − 0.278·15-s + 0.634·17-s − 0.786·19-s + 0.567·23-s − 0.956·25-s + 0.282·27-s + 0.223·29-s − 1.89·31-s + 1.85·33-s + 0.779·37-s + 0.268·39-s − 1.16·41-s − 0.114·43-s + 0.164·45-s − 0.768·47-s − 0.849·51-s + 0.850·53-s − 0.288·55-s + 1.05·57-s − 1.27·59-s + 0.955·61-s − 0.0417·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4685634792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4685634792\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 20.8T + 243T^{2} \) |
| 5 | \( 1 - 11.6T + 3.12e3T^{2} \) |
| 11 | \( 1 + 556.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 122.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 756.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.23e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.44e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.01e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.01e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.49e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.25e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.38e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.16e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.74e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.40e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.77e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.88e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.27e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.11e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.89e3T + 8.58e9T^{2} \) |
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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
−9.880853447491791434087898686654, −8.660878227795355794248855249033, −7.67497551938847134124176090377, −6.83655678053527235327553823412, −5.74457571602255669597686449246, −5.38254837683017200968545414972, −4.38183364743765870993812180686, −2.98492201187461024807313455617, −1.71944092723299531447219436990, −0.32928099524296331973810483529,
0.32928099524296331973810483529, 1.71944092723299531447219436990, 2.98492201187461024807313455617, 4.38183364743765870993812180686, 5.38254837683017200968545414972, 5.74457571602255669597686449246, 6.83655678053527235327553823412, 7.67497551938847134124176090377, 8.660878227795355794248855249033, 9.880853447491791434087898686654
