| L(s) = 1 | − 12·3-s + 54·5-s + 88·7-s − 99·9-s + 418·13-s − 648·15-s − 594·17-s − 836·19-s − 1.05e3·21-s − 4.10e3·23-s − 209·25-s + 4.10e3·27-s + 594·29-s + 4.25e3·31-s + 4.75e3·35-s − 298·37-s − 5.01e3·39-s − 1.72e4·41-s + 1.21e4·43-s − 5.34e3·45-s − 1.29e3·47-s − 9.06e3·49-s + 7.12e3·51-s + 1.94e4·53-s + 1.00e4·57-s − 7.66e3·59-s + 3.47e4·61-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 0.965·5-s + 0.678·7-s − 0.407·9-s + 0.685·13-s − 0.743·15-s − 0.498·17-s − 0.531·19-s − 0.522·21-s − 1.61·23-s − 0.0668·25-s + 1.08·27-s + 0.131·29-s + 0.795·31-s + 0.655·35-s − 0.0357·37-s − 0.528·39-s − 1.60·41-s + 0.997·43-s − 0.393·45-s − 0.0855·47-s − 0.539·49-s + 0.383·51-s + 0.953·53-s + 0.408·57-s − 0.286·59-s + 1.19·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 4 p T + p^{5} T^{2} \) |
| 5 | \( 1 - 54 T + p^{5} T^{2} \) |
| 7 | \( 1 - 88 T + p^{5} T^{2} \) |
| 13 | \( 1 - 418 T + p^{5} T^{2} \) |
| 17 | \( 1 + 594 T + p^{5} T^{2} \) |
| 19 | \( 1 + 44 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 4104 T + p^{5} T^{2} \) |
| 29 | \( 1 - 594 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4256 T + p^{5} T^{2} \) |
| 37 | \( 1 + 298 T + p^{5} T^{2} \) |
| 41 | \( 1 + 17226 T + p^{5} T^{2} \) |
| 43 | \( 1 - 12100 T + p^{5} T^{2} \) |
| 47 | \( 1 + 1296 T + p^{5} T^{2} \) |
| 53 | \( 1 - 19494 T + p^{5} T^{2} \) |
| 59 | \( 1 + 7668 T + p^{5} T^{2} \) |
| 61 | \( 1 - 34738 T + p^{5} T^{2} \) |
| 67 | \( 1 - 21812 T + p^{5} T^{2} \) |
| 71 | \( 1 + 46872 T + p^{5} T^{2} \) |
| 73 | \( 1 + 67562 T + p^{5} T^{2} \) |
| 79 | \( 1 - 76912 T + p^{5} T^{2} \) |
| 83 | \( 1 + 67716 T + p^{5} T^{2} \) |
| 89 | \( 1 - 29754 T + p^{5} T^{2} \) |
| 97 | \( 1 + 122398 T + p^{5} T^{2} \) |
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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
−9.954804251071841470963738848208, −8.777571863634882492423916255724, −8.077673245066842690115029438335, −6.61636027804203576303100699723, −5.97168778396615894239372471009, −5.20032710386346949080640632222, −4.07774993352880939699943348752, −2.45553399370002005608210698952, −1.42264110042154746054236971433, 0,
1.42264110042154746054236971433, 2.45553399370002005608210698952, 4.07774993352880939699943348752, 5.20032710386346949080640632222, 5.97168778396615894239372471009, 6.61636027804203576303100699723, 8.077673245066842690115029438335, 8.777571863634882492423916255724, 9.954804251071841470963738848208
