| L(s) = 1 | − 4·2-s + 27.3·3-s + 16·4-s − 91.1·5-s − 109.·6-s + 45.5·7-s − 64·8-s + 504.·9-s + 364.·10-s − 407.·11-s + 437.·12-s − 421.·13-s − 182.·14-s − 2.49e3·15-s + 256·16-s − 844.·17-s − 2.01e3·18-s − 1.45e3·20-s + 1.24e3·21-s + 1.63e3·22-s − 4.66e3·23-s − 1.74e3·24-s + 5.18e3·25-s + 1.68e3·26-s + 7.14e3·27-s + 729.·28-s + 3.99e3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.75·3-s + 0.5·4-s − 1.63·5-s − 1.24·6-s + 0.351·7-s − 0.353·8-s + 2.07·9-s + 1.15·10-s − 1.01·11-s + 0.876·12-s − 0.691·13-s − 0.248·14-s − 2.85·15-s + 0.250·16-s − 0.708·17-s − 1.46·18-s − 0.815·20-s + 0.616·21-s + 0.718·22-s − 1.83·23-s − 0.620·24-s + 1.65·25-s + 0.488·26-s + 1.88·27-s + 0.175·28-s + 0.882·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.742101567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.742101567\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 27.3T + 243T^{2} \) |
| 5 | \( 1 + 91.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 45.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 407.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 421.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 844.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.66e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.99e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.46e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.25e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.25e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.05e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.32e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.06e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.18e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.65e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.91e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.17e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.48e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.44e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.10e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.30e5T + 8.58e9T^{2} \) |
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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
−9.522245429634451943909857138138, −8.383034535636862394725333641917, −8.032704766421002195638603091911, −7.69551375572221430747265444840, −6.64237931014966548103714912497, −4.69076751532376651477196785640, −3.94609817494237505568619204391, −2.85854339880985455795614221122, −2.18200048842691166351894782278, −0.60271350534296704619816469924,
0.60271350534296704619816469924, 2.18200048842691166351894782278, 2.85854339880985455795614221122, 3.94609817494237505568619204391, 4.69076751532376651477196785640, 6.64237931014966548103714912497, 7.69551375572221430747265444840, 8.032704766421002195638603091911, 8.383034535636862394725333641917, 9.522245429634451943909857138138
