| L(s) = 1 | + 23.5·3-s + 74.2·5-s + 310.·9-s + 424.·11-s + 252.·13-s + 1.74e3·15-s + 1.10e3·17-s − 6.47·19-s + 3.61e3·23-s + 2.39e3·25-s + 1.57e3·27-s − 5.00e3·29-s + 2.82e3·31-s + 9.97e3·33-s − 2.04e3·37-s + 5.93e3·39-s − 9.39e3·41-s − 1.03e4·43-s + 2.30e4·45-s + 1.70e4·47-s + 2.59e4·51-s − 3.95e4·53-s + 3.15e4·55-s − 152.·57-s + 3.39e4·59-s − 2.82e4·61-s + 1.87e4·65-s + ⋯ |
| L(s) = 1 | + 1.50·3-s + 1.32·5-s + 1.27·9-s + 1.05·11-s + 0.413·13-s + 2.00·15-s + 0.926·17-s − 0.00411·19-s + 1.42·23-s + 0.765·25-s + 0.416·27-s − 1.10·29-s + 0.527·31-s + 1.59·33-s − 0.245·37-s + 0.624·39-s − 0.872·41-s − 0.851·43-s + 1.69·45-s + 1.12·47-s + 1.39·51-s − 1.93·53-s + 1.40·55-s − 0.00620·57-s + 1.26·59-s − 0.973·61-s + 0.550·65-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
| \(L(3)\) |
\(\approx\) |
\(6.353811252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.353811252\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 23.5T + 243T^{2} \) |
| 5 | \( 1 - 74.2T + 3.12e3T^{2} \) |
| 11 | \( 1 - 424.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 252.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.10e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 6.47T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.61e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.04e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.70e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.95e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.39e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.55e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.82e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.38e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.88e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.08e5T + 8.58e9T^{2} \) |
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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
−9.355372912020021730241351118574, −8.931459943132264806971307981586, −8.002199276282416304442161762037, −7.00402485784650260756494603427, −6.12633707900141863610938475649, −5.08013532794258773703006378565, −3.74072775402439171627684324660, −3.00390087799909120649706369520, −1.91865051537295262688619234919, −1.21947282295432763668721103644,
1.21947282295432763668721103644, 1.91865051537295262688619234919, 3.00390087799909120649706369520, 3.74072775402439171627684324660, 5.08013532794258773703006378565, 6.12633707900141863610938475649, 7.00402485784650260756494603427, 8.002199276282416304442161762037, 8.931459943132264806971307981586, 9.355372912020021730241351118574
