| L(s) = 1 | − 2-s + 4-s − 2.18·5-s − 2.20·7-s − 8-s + 2.18·10-s − 3.05·11-s + 5.61·13-s + 2.20·14-s + 16-s − 3.13·17-s − 6.33·19-s − 2.18·20-s + 3.05·22-s + 8.51·23-s − 0.208·25-s − 5.61·26-s − 2.20·28-s − 9.65·29-s + 0.683·31-s − 32-s + 3.13·34-s + 4.81·35-s + 4.31·37-s + 6.33·38-s + 2.18·40-s − 9.53·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.978·5-s − 0.831·7-s − 0.353·8-s + 0.692·10-s − 0.921·11-s + 1.55·13-s + 0.588·14-s + 0.250·16-s − 0.761·17-s − 1.45·19-s − 0.489·20-s + 0.651·22-s + 1.77·23-s − 0.0417·25-s − 1.10·26-s − 0.415·28-s − 1.79·29-s + 0.122·31-s − 0.176·32-s + 0.538·34-s + 0.814·35-s + 0.708·37-s + 1.02·38-s + 0.346·40-s − 1.48·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5305811652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5305811652\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 + 2.18T + 5T^{2} \) |
| 7 | \( 1 + 2.20T + 7T^{2} \) |
| 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 - 5.61T + 13T^{2} \) |
| 17 | \( 1 + 3.13T + 17T^{2} \) |
| 19 | \( 1 + 6.33T + 19T^{2} \) |
| 23 | \( 1 - 8.51T + 23T^{2} \) |
| 29 | \( 1 + 9.65T + 29T^{2} \) |
| 31 | \( 1 - 0.683T + 31T^{2} \) |
| 37 | \( 1 - 4.31T + 37T^{2} \) |
| 41 | \( 1 + 9.53T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 + 7.10T + 47T^{2} \) |
| 53 | \( 1 - 2.20T + 53T^{2} \) |
| 59 | \( 1 + 1.58T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 - 7.04T + 67T^{2} \) |
| 71 | \( 1 - 6.65T + 71T^{2} \) |
| 73 | \( 1 + 8.06T + 73T^{2} \) |
| 79 | \( 1 - 3.90T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 9.61T + 89T^{2} \) |
| 97 | \( 1 + 3.27T + 97T^{2} \) |
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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
−8.508278649434738609750774158596, −7.84237066595816839960809548465, −7.04569410931586873608561165334, −6.44399436780333668927164405734, −5.67865296100465004894236216515, −4.54412917646754247047070124146, −3.67057132885269572661276924746, −3.01975942221540378859808263982, −1.86510385555534692843195022739, −0.44695084338139164166953540236,
0.44695084338139164166953540236, 1.86510385555534692843195022739, 3.01975942221540378859808263982, 3.67057132885269572661276924746, 4.54412917646754247047070124146, 5.67865296100465004894236216515, 6.44399436780333668927164405734, 7.04569410931586873608561165334, 7.84237066595816839960809548465, 8.508278649434738609750774158596
