| L(s) = 1 | − 2.23·2-s + 1.62·3-s + 2.97·4-s + 2.06·5-s − 3.62·6-s + 3.32·7-s − 2.17·8-s − 0.353·9-s − 4.61·10-s − 2.00·11-s + 4.83·12-s − 4.84·13-s − 7.42·14-s + 3.36·15-s − 1.10·16-s − 1.62·17-s + 0.789·18-s − 7.82·19-s + 6.15·20-s + 5.41·21-s + 4.46·22-s + 1.63·23-s − 3.53·24-s − 0.720·25-s + 10.8·26-s − 5.45·27-s + 9.89·28-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 0.939·3-s + 1.48·4-s + 0.925·5-s − 1.48·6-s + 1.25·7-s − 0.767·8-s − 0.117·9-s − 1.45·10-s − 0.604·11-s + 1.39·12-s − 1.34·13-s − 1.98·14-s + 0.868·15-s − 0.276·16-s − 0.393·17-s + 0.186·18-s − 1.79·19-s + 1.37·20-s + 1.18·21-s + 0.952·22-s + 0.341·23-s − 0.720·24-s − 0.144·25-s + 2.12·26-s − 1.04·27-s + 1.87·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 - 1.62T + 3T^{2} \) |
| 5 | \( 1 - 2.06T + 5T^{2} \) |
| 7 | \( 1 - 3.32T + 7T^{2} \) |
| 11 | \( 1 + 2.00T + 11T^{2} \) |
| 13 | \( 1 + 4.84T + 13T^{2} \) |
| 17 | \( 1 + 1.62T + 17T^{2} \) |
| 19 | \( 1 + 7.82T + 19T^{2} \) |
| 23 | \( 1 - 1.63T + 23T^{2} \) |
| 31 | \( 1 - 6.32T + 31T^{2} \) |
| 37 | \( 1 - 5.26T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 2.54T + 43T^{2} \) |
| 47 | \( 1 + 0.424T + 47T^{2} \) |
| 53 | \( 1 + 6.57T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 8.32T + 61T^{2} \) |
| 67 | \( 1 + 5.61T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 1.51T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 9.67T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.248935525325239724557153023281, −7.75124454063239380420933085183, −6.99242873280078820495066080394, −6.09787640684786588861635787371, −5.06082350092234386787201405254, −4.34656268678035115323436455528, −2.67688713863515323388385989004, −2.25681503206110624395369362948, −1.59563464841583236358920066320, 0,
1.59563464841583236358920066320, 2.25681503206110624395369362948, 2.67688713863515323388385989004, 4.34656268678035115323436455528, 5.06082350092234386787201405254, 6.09787640684786588861635787371, 6.99242873280078820495066080394, 7.75124454063239380420933085183, 8.248935525325239724557153023281
