| L(s) = 1 | + 8.69e10·2-s + 2.84e21·4-s + 2.65e30·7-s − 1.63e32·8-s + 2.25e34·9-s − 5.36e37·11-s + 2.30e41·14-s − 2.76e43·16-s + 1.95e45·18-s − 4.66e48·22-s − 6.95e48·23-s + 2.11e50·25-s + 7.53e51·28-s + 5.63e52·29-s − 1.63e54·32-s + 6.39e55·36-s + 4.82e56·37-s + 7.29e57·43-s − 1.52e59·44-s − 6.04e59·46-s + 7.03e60·49-s + 1.84e61·50-s − 1.06e62·53-s − 4.33e62·56-s + 4.89e63·58-s + 5.97e64·63-s − 1.13e64·64-s + ⋯ |
| L(s) = 1 | + 1.26·2-s + 0.601·4-s + 7-s − 0.504·8-s + 9-s − 1.73·11-s + 1.26·14-s − 1.23·16-s + 1.26·18-s − 2.19·22-s − 0.660·23-s + 25-s + 0.601·28-s + 1.27·29-s − 1.06·32-s + 0.601·36-s + 1.68·37-s + 0.114·43-s − 1.04·44-s − 0.835·46-s + 49-s + 1.26·50-s − 0.902·53-s − 0.504·56-s + 1.60·58-s + 63-s − 0.107·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(73-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+36) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{73}{2})\) |
\(\approx\) |
\(4.753168720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.753168720\) |
| \(L(37)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - p^{36} T \) |
| good | 2 | \( 1 - 86965052897 T + p^{72} T^{2} \) |
| 3 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 5 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 11 | \( 1 + \)\(53\!\cdots\!78\)\( T + p^{72} T^{2} \) |
| 13 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 17 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 19 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 23 | \( 1 + \)\(69\!\cdots\!78\)\( T + p^{72} T^{2} \) |
| 29 | \( 1 - \)\(56\!\cdots\!42\)\( T + p^{72} T^{2} \) |
| 31 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 37 | \( 1 - \)\(48\!\cdots\!82\)\( T + p^{72} T^{2} \) |
| 41 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 43 | \( 1 - \)\(72\!\cdots\!02\)\( T + p^{72} T^{2} \) |
| 47 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 53 | \( 1 + \)\(10\!\cdots\!58\)\( T + p^{72} T^{2} \) |
| 59 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 61 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 67 | \( 1 - \)\(51\!\cdots\!62\)\( T + p^{72} T^{2} \) |
| 71 | \( 1 + \)\(47\!\cdots\!58\)\( T + p^{72} T^{2} \) |
| 73 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 79 | \( 1 + \)\(24\!\cdots\!58\)\( T + p^{72} T^{2} \) |
| 83 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 89 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 97 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
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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
−11.14361318316090511087960433572, −10.04205985280867893623719616930, −8.398071914474330980154897174421, −7.35142261650743127226281193932, −5.97083665415305217170572877931, −4.86710809093647910989838746685, −4.43692412097376782189500977863, −3.02124646122550062676593310106, −2.09630435475363123786102040514, −0.72594863721078773032705296147,
0.72594863721078773032705296147, 2.09630435475363123786102040514, 3.02124646122550062676593310106, 4.43692412097376782189500977863, 4.86710809093647910989838746685, 5.97083665415305217170572877931, 7.35142261650743127226281193932, 8.398071914474330980154897174421, 10.04205985280867893623719616930, 11.14361318316090511087960433572
