| L(s) = 1 | − 2.56·2-s + 3-s + 4.56·4-s − 0.561·5-s − 2.56·6-s + 3.56·7-s − 6.56·8-s + 9-s + 1.43·10-s + 2·11-s + 4.56·12-s − 9.12·14-s − 0.561·15-s + 7.68·16-s + 2.56·17-s − 2.56·18-s + 1.12·19-s − 2.56·20-s + 3.56·21-s − 5.12·22-s + 2·23-s − 6.56·24-s − 4.68·25-s + 27-s + 16.2·28-s − 5.68·29-s + 1.43·30-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 0.577·3-s + 2.28·4-s − 0.251·5-s − 1.04·6-s + 1.34·7-s − 2.31·8-s + 0.333·9-s + 0.454·10-s + 0.603·11-s + 1.31·12-s − 2.43·14-s − 0.144·15-s + 1.92·16-s + 0.621·17-s − 0.603·18-s + 0.257·19-s − 0.572·20-s + 0.777·21-s − 1.09·22-s + 0.417·23-s − 1.33·24-s − 0.936·25-s + 0.192·27-s + 3.07·28-s − 1.05·29-s + 0.262·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8982542050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8982542050\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 5 | \( 1 + 0.561T + 5T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 - 1.56T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 + 2.56T + 41T^{2} \) |
| 43 | \( 1 - 0.438T + 43T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 0.438T + 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 - 1.87T + 73T^{2} \) |
| 79 | \( 1 - 9.56T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 4.43T + 97T^{2} \) |
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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
−10.68832318616732055991105685269, −9.846403248930172989474048516157, −8.992562004097801047001551216995, −8.327150364814828599349628820907, −7.64231171039695604004499274615, −6.95723307936731005989370733517, −5.48589071181208357151227696095, −3.86824313279929653220187459391, −2.27452445155264935855313400762, −1.21501910271491189756816325506,
1.21501910271491189756816325506, 2.27452445155264935855313400762, 3.86824313279929653220187459391, 5.48589071181208357151227696095, 6.95723307936731005989370733517, 7.64231171039695604004499274615, 8.327150364814828599349628820907, 8.992562004097801047001551216995, 9.846403248930172989474048516157, 10.68832318616732055991105685269
