| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s + 9-s − 2·10-s − 2·12-s − 3·13-s + 15-s − 4·16-s − 4·17-s + 2·18-s + 19-s − 2·20-s − 4·23-s + 25-s − 6·26-s − 27-s + 8·29-s + 2·30-s − 31-s − 8·32-s − 8·34-s + 2·36-s + 7·37-s + 2·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 1/3·9-s − 0.632·10-s − 0.577·12-s − 0.832·13-s + 0.258·15-s − 16-s − 0.970·17-s + 0.471·18-s + 0.229·19-s − 0.447·20-s − 0.834·23-s + 1/5·25-s − 1.17·26-s − 0.192·27-s + 1.48·29-s + 0.365·30-s − 0.179·31-s − 1.41·32-s − 1.37·34-s + 1/3·36-s + 1.15·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.04365882526187, −13.59282517768704, −13.17176079956834, −12.59741816682858, −12.19593574966103, −11.83092182674195, −11.42052354212597, −10.89928354157084, −10.28329934294972, −9.787297963789879, −9.164097746279994, −8.551209881278475, −7.948098724064664, −7.350838112180036, −6.706866907412219, −6.432861453837772, −5.829649851499875, −5.194776771386875, −4.709251817296001, −4.410833351158110, −3.764571398267851, −3.163450654875430, −2.470283364383200, −1.957097365500609, −0.7980892570866913, 0,
0.7980892570866913, 1.957097365500609, 2.470283364383200, 3.163450654875430, 3.764571398267851, 4.410833351158110, 4.709251817296001, 5.194776771386875, 5.829649851499875, 6.432861453837772, 6.706866907412219, 7.350838112180036, 7.948098724064664, 8.551209881278475, 9.164097746279994, 9.787297963789879, 10.28329934294972, 10.89928354157084, 11.42052354212597, 11.83092182674195, 12.19593574966103, 12.59741816682858, 13.17176079956834, 13.59282517768704, 14.04365882526187
