| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s − 3·7-s + 9-s − 2·12-s − 6·14-s − 4·16-s − 8·17-s + 2·18-s + 19-s + 3·21-s + 6·23-s − 5·25-s − 27-s − 6·28-s + 2·29-s + 9·31-s − 8·32-s − 16·34-s + 2·36-s + 9·37-s + 2·38-s − 6·41-s + 6·42-s − 4·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s − 1.13·7-s + 1/3·9-s − 0.577·12-s − 1.60·14-s − 16-s − 1.94·17-s + 0.471·18-s + 0.229·19-s + 0.654·21-s + 1.25·23-s − 25-s − 0.192·27-s − 1.13·28-s + 0.371·29-s + 1.61·31-s − 1.41·32-s − 2.74·34-s + 1/3·36-s + 1.47·37-s + 0.324·38-s − 0.937·41-s + 0.925·42-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.299273161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.299273161\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.13059843404222, −13.48714873819108, −13.30300547548900, −12.95029302956283, −12.43886233164835, −11.79087382975887, −11.40674872012735, −11.07116144608949, −10.28210456561722, −9.678188665569130, −9.359560721147229, −8.599265074872470, −8.047800146429104, −6.988158659535306, −6.770309269107213, −6.283049251511957, −5.916088609055085, −5.094675444693876, −4.650488266053763, −4.240447049075811, −3.500834194808735, −2.903765137918238, −2.456291993181612, −1.461824422287968, −0.3052201250228582,
0.3052201250228582, 1.461824422287968, 2.456291993181612, 2.903765137918238, 3.500834194808735, 4.240447049075811, 4.650488266053763, 5.094675444693876, 5.916088609055085, 6.283049251511957, 6.770309269107213, 6.988158659535306, 8.047800146429104, 8.599265074872470, 9.359560721147229, 9.678188665569130, 10.28210456561722, 11.07116144608949, 11.40674872012735, 11.79087382975887, 12.43886233164835, 12.95029302956283, 13.30300547548900, 13.48714873819108, 14.13059843404222
