| L(s) = 1 | − 1.42·2-s − 3-s + 0.0388·4-s + 5-s + 1.42·6-s + 1.25·7-s + 2.80·8-s + 9-s − 1.42·10-s + 0.401·11-s − 0.0388·12-s − 5.80·13-s − 1.78·14-s − 15-s − 4.07·16-s + 3.55·17-s − 1.42·18-s − 6.76·19-s + 0.0388·20-s − 1.25·21-s − 0.573·22-s − 2.80·24-s + 25-s + 8.28·26-s − 27-s + 0.0486·28-s − 5.50·29-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 0.577·3-s + 0.0194·4-s + 0.447·5-s + 0.582·6-s + 0.473·7-s + 0.990·8-s + 0.333·9-s − 0.451·10-s + 0.121·11-s − 0.0112·12-s − 1.60·13-s − 0.478·14-s − 0.258·15-s − 1.01·16-s + 0.861·17-s − 0.336·18-s − 1.55·19-s + 0.00867·20-s − 0.273·21-s − 0.122·22-s − 0.571·24-s + 0.200·25-s + 1.62·26-s − 0.192·27-s + 0.00919·28-s − 1.02·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5053710881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5053710881\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.42T + 2T^{2} \) |
| 7 | \( 1 - 1.25T + 7T^{2} \) |
| 11 | \( 1 - 0.401T + 11T^{2} \) |
| 13 | \( 1 + 5.80T + 13T^{2} \) |
| 17 | \( 1 - 3.55T + 17T^{2} \) |
| 19 | \( 1 + 6.76T + 19T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 + 11.0T + 31T^{2} \) |
| 37 | \( 1 - 4.78T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 9.81T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 1.43T + 53T^{2} \) |
| 59 | \( 1 - 9.68T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 7.14T + 71T^{2} \) |
| 73 | \( 1 + 3.47T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 3.32T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 1.15T + 97T^{2} \) |
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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
−7.83203215980063556080824842878, −7.28673914700486294726908699629, −6.69806881994979169516726978412, −5.66373540131936271394550886465, −5.10886908959578314001094272032, −4.49103775386192095552314432801, −3.57798257254983859864277501114, −2.14887508749957217177796625554, −1.71219041091856878821976134017, −0.42258388389879229630174528961,
0.42258388389879229630174528961, 1.71219041091856878821976134017, 2.14887508749957217177796625554, 3.57798257254983859864277501114, 4.49103775386192095552314432801, 5.10886908959578314001094272032, 5.66373540131936271394550886465, 6.69806881994979169516726978412, 7.28673914700486294726908699629, 7.83203215980063556080824842878
