| L(s) = 1 | − 1.41·2-s + 1.00·4-s + 1.84·5-s + 9-s − 2.61·10-s + 0.765·11-s − 0.999·16-s − 17-s − 1.41·18-s + 1.84·20-s − 1.08·22-s − 1.84·23-s + 2.41·25-s − 0.765·29-s − 0.765·31-s + 1.41·32-s + 1.41·34-s + 1.00·36-s − 1.84·41-s + 0.765·44-s + 1.84·45-s + 2.61·46-s − 47-s + 49-s − 3.41·50-s + 1.41·55-s + 1.08·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.00·4-s + 1.84·5-s + 9-s − 2.61·10-s + 0.765·11-s − 0.999·16-s − 17-s − 1.41·18-s + 1.84·20-s − 1.08·22-s − 1.84·23-s + 2.41·25-s − 0.765·29-s − 0.765·31-s + 1.41·32-s + 1.41·34-s + 1.00·36-s − 1.84·41-s + 0.765·44-s + 1.84·45-s + 2.61·46-s − 47-s + 49-s − 3.41·50-s + 1.41·55-s + 1.08·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7019531681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7019531681\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 1.41T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.84T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 0.765T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.84T + T^{2} \) |
| 29 | \( 1 + 0.765T + T^{2} \) |
| 31 | \( 1 + 0.765T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.84T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.765T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.14064795174349330118062943248, −9.612807905988525046426979837388, −9.078286563532982412464251826877, −8.185903378605004678847052633019, −6.95202882038181500529883090318, −6.49664325122377342082172518890, −5.36033789114543979090999479686, −4.09745807698964468747766934923, −2.11678113221751284338357849876, −1.58425295725740731501318166152,
1.58425295725740731501318166152, 2.11678113221751284338357849876, 4.09745807698964468747766934923, 5.36033789114543979090999479686, 6.49664325122377342082172518890, 6.95202882038181500529883090318, 8.185903378605004678847052633019, 9.078286563532982412464251826877, 9.612807905988525046426979837388, 10.14064795174349330118062943248
