| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s − 3·5-s − 4·6-s + 9-s − 6·10-s + 4·11-s − 4·12-s + 6·13-s + 6·15-s − 4·16-s + 7·17-s + 2·18-s − 19-s − 6·20-s + 8·22-s + 3·23-s + 4·25-s + 12·26-s + 4·27-s + 12·30-s − 8·32-s − 8·33-s + 14·34-s + 2·36-s − 2·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s − 1.34·5-s − 1.63·6-s + 1/3·9-s − 1.89·10-s + 1.20·11-s − 1.15·12-s + 1.66·13-s + 1.54·15-s − 16-s + 1.69·17-s + 0.471·18-s − 0.229·19-s − 1.34·20-s + 1.70·22-s + 0.625·23-s + 4/5·25-s + 2.35·26-s + 0.769·27-s + 2.19·30-s − 1.41·32-s − 1.39·33-s + 2.40·34-s + 1/3·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.875793980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.875793980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−10.64416687445000343603953164841, −9.169422038731926349646733763033, −8.257216412779562315668451423776, −7.13470928968254736566438211223, −6.25883493919142594991046086243, −5.71582158035303955673922475384, −4.70997962935936595905677293224, −3.83386548839045730562065401010, −3.32672471138068793751574139576, −0.979388952363395344123223860435,
0.979388952363395344123223860435, 3.32672471138068793751574139576, 3.83386548839045730562065401010, 4.70997962935936595905677293224, 5.71582158035303955673922475384, 6.25883493919142594991046086243, 7.13470928968254736566438211223, 8.257216412779562315668451423776, 9.169422038731926349646733763033, 10.64416687445000343603953164841
