| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 8-s + 9-s − 2·12-s − 4·13-s + 16-s − 6·17-s + 18-s − 2·24-s − 5·25-s − 4·26-s + 4·27-s + 6·29-s − 4·31-s + 32-s − 6·34-s + 36-s − 2·37-s + 8·39-s + 6·41-s + 8·43-s + 12·47-s − 2·48-s − 5·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.577·12-s − 1.10·13-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.408·24-s − 25-s − 0.784·26-s + 0.769·27-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 1.28·39-s + 0.937·41-s + 1.21·43-s + 1.75·47-s − 0.288·48-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.030824817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030824817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.99728502401873, −14.25983054662366, −13.94531513785829, −13.35276887883669, −12.56731102069754, −12.37388464531861, −11.87873551398178, −11.26962720487718, −10.81783723073702, −10.49521792945982, −9.664486576660142, −9.156955853739614, −8.455785059232210, −7.593989850498145, −7.226035651822059, −6.498461763115004, −6.106204737633401, −5.506725255735893, −4.963388069368697, −4.379223996486467, −3.957921222641953, −2.770787167079069, −2.428566160315683, −1.438665858952500, −0.3562927379342628,
0.3562927379342628, 1.438665858952500, 2.428566160315683, 2.770787167079069, 3.957921222641953, 4.379223996486467, 4.963388069368697, 5.506725255735893, 6.106204737633401, 6.498461763115004, 7.226035651822059, 7.593989850498145, 8.455785059232210, 9.156955853739614, 9.664486576660142, 10.49521792945982, 10.81783723073702, 11.26962720487718, 11.87873551398178, 12.37388464531861, 12.56731102069754, 13.35276887883669, 13.94531513785829, 14.25983054662366, 14.99728502401873
