| L(s) = 1 | + 2.08·2-s + 3.36·3-s + 2.36·4-s − 0.0546·5-s + 7.02·6-s − 2.53·7-s + 0.756·8-s + 8.30·9-s − 0.114·10-s − 11-s + 7.94·12-s − 5.23·13-s − 5.29·14-s − 0.183·15-s − 3.14·16-s − 1.86·17-s + 17.3·18-s + 5.90·19-s − 0.129·20-s − 8.51·21-s − 2.08·22-s − 3.02·23-s + 2.54·24-s − 4.99·25-s − 10.9·26-s + 17.8·27-s − 5.98·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.94·3-s + 1.18·4-s − 0.0244·5-s + 2.86·6-s − 0.957·7-s + 0.267·8-s + 2.76·9-s − 0.0360·10-s − 0.301·11-s + 2.29·12-s − 1.45·13-s − 1.41·14-s − 0.0474·15-s − 0.786·16-s − 0.452·17-s + 4.08·18-s + 1.35·19-s − 0.0288·20-s − 1.85·21-s − 0.445·22-s − 0.630·23-s + 0.519·24-s − 0.999·25-s − 2.14·26-s + 3.43·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.036237183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.036237183\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 3 | \( 1 - 3.36T + 3T^{2} \) |
| 5 | \( 1 + 0.0546T + 5T^{2} \) |
| 7 | \( 1 + 2.53T + 7T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 + 1.86T + 17T^{2} \) |
| 19 | \( 1 - 5.90T + 19T^{2} \) |
| 23 | \( 1 + 3.02T + 23T^{2} \) |
| 29 | \( 1 + 5.01T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 + 6.62T + 37T^{2} \) |
| 41 | \( 1 - 6.90T + 41T^{2} \) |
| 43 | \( 1 - 8.34T + 43T^{2} \) |
| 47 | \( 1 - 7.83T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 7.54T + 59T^{2} \) |
| 67 | \( 1 - 1.62T + 67T^{2} \) |
| 71 | \( 1 + 7.43T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 8.45T + 79T^{2} \) |
| 83 | \( 1 - 0.244T + 83T^{2} \) |
| 89 | \( 1 - 5.48T + 89T^{2} \) |
| 97 | \( 1 + 1.77T + 97T^{2} \) |
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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
−10.20048653271584711178841034376, −9.612950626350845174099180167182, −8.876465858529497053120227435892, −7.59102661362734520295876902515, −7.15195443995639498041158954686, −5.88163280411114386319800255853, −4.60616973373228835916426173859, −3.82611352434173211175393480494, −2.90799671552016498857960697548, −2.33342680754643133064458235922,
2.33342680754643133064458235922, 2.90799671552016498857960697548, 3.82611352434173211175393480494, 4.60616973373228835916426173859, 5.88163280411114386319800255853, 7.15195443995639498041158954686, 7.59102661362734520295876902515, 8.876465858529497053120227435892, 9.612950626350845174099180167182, 10.20048653271584711178841034376
