| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3425291637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3425291637\) |
| \(L(1)\) |
|
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
−10.58279108889080707195992803554, −9.500232934364953692314729484149, −8.635043526895702006734981567288, −8.048299351295963846142896233099, −7.25385161510358371747494107528, −6.82861976462304510514837837719, −4.81802217054357925804148446929, −4.16927173793250938157856854686, −2.73541741301945994096599588112, −0.889799171181315019954404780631,
0.889799171181315019954404780631, 2.73541741301945994096599588112, 4.16927173793250938157856854686, 4.81802217054357925804148446929, 6.82861976462304510514837837719, 7.25385161510358371747494107528, 8.048299351295963846142896233099, 8.635043526895702006734981567288, 9.500232934364953692314729484149, 10.58279108889080707195992803554
