| L(s) = 1 | + 2·2-s + 2·4-s − 3·5-s − 3·9-s − 6·10-s + 6·11-s − 4·16-s − 4·17-s − 6·18-s + 5·19-s − 6·20-s + 12·22-s + 3·23-s + 4·25-s − 5·29-s − 3·31-s − 8·32-s − 8·34-s − 6·36-s + 4·37-s + 10·38-s − 6·41-s − 43-s + 12·44-s + 9·45-s + 6·46-s + 7·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.34·5-s − 9-s − 1.89·10-s + 1.80·11-s − 16-s − 0.970·17-s − 1.41·18-s + 1.14·19-s − 1.34·20-s + 2.55·22-s + 0.625·23-s + 4/5·25-s − 0.928·29-s − 0.538·31-s − 1.41·32-s − 1.37·34-s − 36-s + 0.657·37-s + 1.62·38-s − 0.937·41-s − 0.152·43-s + 1.80·44-s + 1.34·45-s + 0.884·46-s + 1.02·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.799788347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.799788347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.57740679946665025654456265521, −6.90678100068040939590507024634, −6.38026602914856538020136886750, −5.57634080836583272363145209435, −4.89731593269989771358095794547, −4.14522542139967607027503862649, −3.61956902475371613886505030678, −3.18335929774170366064894969897, −2.07697976816031338635613740258, −0.64616466383857911844066535116,
0.64616466383857911844066535116, 2.07697976816031338635613740258, 3.18335929774170366064894969897, 3.61956902475371613886505030678, 4.14522542139967607027503862649, 4.89731593269989771358095794547, 5.57634080836583272363145209435, 6.38026602914856538020136886750, 6.90678100068040939590507024634, 7.57740679946665025654456265521
