| L(s) = 1 | + 2.79e11·2-s + 4.00e22·4-s + 2.13e26·5-s + 8.63e31·7-s + 6.46e32·8-s + 5.96e37·10-s − 5.98e36·11-s + 7.00e41·13-s + 2.41e43·14-s − 1.33e45·16-s + 1.90e46·17-s − 1.27e48·19-s + 8.56e48·20-s − 1.66e48·22-s + 2.12e51·23-s + 1.91e52·25-s + 1.95e53·26-s + 3.46e54·28-s − 3.23e54·29-s − 1.75e55·31-s − 3.96e56·32-s + 5.30e57·34-s + 1.84e58·35-s + 1.02e59·37-s − 3.56e59·38-s + 1.38e59·40-s + 3.76e60·41-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.06·4-s + 1.31·5-s + 1.75·7-s + 0.0880·8-s + 1.88·10-s − 0.00530·11-s + 1.18·13-s + 2.52·14-s − 0.934·16-s + 1.37·17-s − 1.42·19-s + 1.39·20-s − 0.00761·22-s + 1.83·23-s + 0.725·25-s + 1.69·26-s + 1.86·28-s − 0.467·29-s − 0.208·31-s − 1.43·32-s + 1.96·34-s + 2.31·35-s + 1.59·37-s − 2.04·38-s + 0.115·40-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(76-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+75/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(38)\) |
\(\approx\) |
\(10.77943323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.77943323\) |
| \(L(\frac{77}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 2.79e11T + 3.77e22T^{2} \) |
| 5 | \( 1 - 2.13e26T + 2.64e52T^{2} \) |
| 7 | \( 1 - 8.63e31T + 2.41e63T^{2} \) |
| 11 | \( 1 + 5.98e36T + 1.27e78T^{2} \) |
| 13 | \( 1 - 7.00e41T + 3.51e83T^{2} \) |
| 17 | \( 1 - 1.90e46T + 1.92e92T^{2} \) |
| 19 | \( 1 + 1.27e48T + 8.06e95T^{2} \) |
| 23 | \( 1 - 2.12e51T + 1.34e102T^{2} \) |
| 29 | \( 1 + 3.23e54T + 4.78e109T^{2} \) |
| 31 | \( 1 + 1.75e55T + 7.11e111T^{2} \) |
| 37 | \( 1 - 1.02e59T + 4.12e117T^{2} \) |
| 41 | \( 1 - 3.76e60T + 9.09e120T^{2} \) |
| 43 | \( 1 + 2.99e61T + 3.23e122T^{2} \) |
| 47 | \( 1 + 3.43e62T + 2.55e125T^{2} \) |
| 53 | \( 1 + 3.26e64T + 2.09e129T^{2} \) |
| 59 | \( 1 - 2.88e66T + 6.51e132T^{2} \) |
| 61 | \( 1 + 2.79e66T + 7.93e133T^{2} \) |
| 67 | \( 1 + 9.66e67T + 9.02e136T^{2} \) |
| 71 | \( 1 - 5.62e68T + 6.98e138T^{2} \) |
| 73 | \( 1 + 2.25e69T + 5.61e139T^{2} \) |
| 79 | \( 1 - 6.48e70T + 2.09e142T^{2} \) |
| 83 | \( 1 - 7.66e71T + 8.52e143T^{2} \) |
| 89 | \( 1 + 6.35e72T + 1.60e146T^{2} \) |
| 97 | \( 1 - 7.22e73T + 1.01e149T^{2} \) |
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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
−10.73385032064677580467407057730, −9.187989587488615018518665455558, −8.069770142665011795114039598942, −6.53568790598619188137945242691, −5.63140540354172542734878327333, −5.02664307721150157049956761604, −4.04994764606554350382954140968, −2.81770638979542060522528679376, −1.81833723044512388978465122973, −1.11616106345223523730923606076,
1.11616106345223523730923606076, 1.81833723044512388978465122973, 2.81770638979542060522528679376, 4.04994764606554350382954140968, 5.02664307721150157049956761604, 5.63140540354172542734878327333, 6.53568790598619188137945242691, 8.069770142665011795114039598942, 9.187989587488615018518665455558, 10.73385032064677580467407057730
