| L(s) = 1 | − 25.2·3-s + 55.5·5-s − 200.·7-s + 395.·9-s − 121·11-s − 727.·13-s − 1.40e3·15-s − 1.10e3·17-s + 1.66e3·19-s + 5.07e3·21-s − 2.98e3·23-s − 40.0·25-s − 3.85e3·27-s + 1.55e3·29-s − 9.51e3·31-s + 3.05e3·33-s − 1.11e4·35-s + 9.43e3·37-s + 1.83e4·39-s + 7.37e3·41-s − 8.52e3·43-s + 2.19e4·45-s − 3.00e4·47-s + 2.35e4·49-s + 2.80e4·51-s − 2.39e4·53-s − 6.72e3·55-s + ⋯ |
| L(s) = 1 | − 1.62·3-s + 0.993·5-s − 1.54·7-s + 1.62·9-s − 0.301·11-s − 1.19·13-s − 1.61·15-s − 0.930·17-s + 1.05·19-s + 2.51·21-s − 1.17·23-s − 0.0128·25-s − 1.01·27-s + 0.342·29-s − 1.77·31-s + 0.488·33-s − 1.53·35-s + 1.13·37-s + 1.93·39-s + 0.684·41-s − 0.703·43-s + 1.61·45-s − 1.98·47-s + 1.39·49-s + 1.50·51-s − 1.17·53-s − 0.299·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1969599271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1969599271\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 + 25.2T + 243T^{2} \) |
| 5 | \( 1 - 55.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 200.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 727.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.10e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.66e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.98e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.43e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.52e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.39e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 6.96e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.90e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.39e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.88e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.15e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.51e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.37e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.91e3T + 8.58e9T^{2} \) |
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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
−9.720630562123374379559436087140, −9.408304952579798658649605854244, −7.59498329118134936485904339990, −6.67946315678667231719971175892, −6.08481685117465706562125065651, −5.44650951679884388796027416252, −4.48529607378281089115996878437, −3.01859498039630301095370983237, −1.76665569747907293649432819174, −0.21695042599729016595842671508,
0.21695042599729016595842671508, 1.76665569747907293649432819174, 3.01859498039630301095370983237, 4.48529607378281089115996878437, 5.44650951679884388796027416252, 6.08481685117465706562125065651, 6.67946315678667231719971175892, 7.59498329118134936485904339990, 9.408304952579798658649605854244, 9.720630562123374379559436087140
