| L(s) = 1 | − 1.17·2-s − 0.618·4-s + 0.0159·5-s − 0.631·7-s + 3.07·8-s − 0.0187·10-s − 2.39·11-s − 6.35·13-s + 0.742·14-s − 2.37·16-s + 0.874·17-s + 19-s − 0.00985·20-s + 2.80·22-s − 8.15·23-s − 4.99·25-s + 7.46·26-s + 0.391·28-s − 7.32·29-s − 0.949·31-s − 3.35·32-s − 1.02·34-s − 0.0100·35-s − 4.80·37-s − 1.17·38-s + 0.0489·40-s − 2.52·41-s + ⋯ |
| L(s) = 1 | − 0.830·2-s − 0.309·4-s + 0.00711·5-s − 0.238·7-s + 1.08·8-s − 0.00591·10-s − 0.720·11-s − 1.76·13-s + 0.198·14-s − 0.594·16-s + 0.212·17-s + 0.229·19-s − 0.00220·20-s + 0.598·22-s − 1.70·23-s − 0.999·25-s + 1.46·26-s + 0.0739·28-s − 1.36·29-s − 0.170·31-s − 0.593·32-s − 0.176·34-s − 0.00169·35-s − 0.789·37-s − 0.190·38-s + 0.00774·40-s − 0.394·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1959973636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1959973636\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 1.17T + 2T^{2} \) |
| 5 | \( 1 - 0.0159T + 5T^{2} \) |
| 7 | \( 1 + 0.631T + 7T^{2} \) |
| 11 | \( 1 + 2.39T + 11T^{2} \) |
| 13 | \( 1 + 6.35T + 13T^{2} \) |
| 17 | \( 1 - 0.874T + 17T^{2} \) |
| 23 | \( 1 + 8.15T + 23T^{2} \) |
| 29 | \( 1 + 7.32T + 29T^{2} \) |
| 31 | \( 1 + 0.949T + 31T^{2} \) |
| 37 | \( 1 + 4.80T + 37T^{2} \) |
| 41 | \( 1 + 2.52T + 41T^{2} \) |
| 43 | \( 1 - 2.63T + 43T^{2} \) |
| 53 | \( 1 + 4.32T + 53T^{2} \) |
| 59 | \( 1 + 4.56T + 59T^{2} \) |
| 61 | \( 1 - 9.48T + 61T^{2} \) |
| 67 | \( 1 - 9.91T + 67T^{2} \) |
| 71 | \( 1 + 3.44T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 + 3.15T + 79T^{2} \) |
| 83 | \( 1 + 5.55T + 83T^{2} \) |
| 89 | \( 1 + 8.93T + 89T^{2} \) |
| 97 | \( 1 + 4.30T + 97T^{2} \) |
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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
−7.79333111241852712683114887623, −7.50243063601855194013604643301, −6.66880807393820930322710760413, −5.57846754364954229873995248806, −5.16246454355192174816234452570, −4.28016149633026900127938127116, −3.55663462805800592557355848640, −2.38508918015545272578579532301, −1.75632276232079511894405496682, −0.23948143507581450751306235544,
0.23948143507581450751306235544, 1.75632276232079511894405496682, 2.38508918015545272578579532301, 3.55663462805800592557355848640, 4.28016149633026900127938127116, 5.16246454355192174816234452570, 5.57846754364954229873995248806, 6.66880807393820930322710760413, 7.50243063601855194013604643301, 7.79333111241852712683114887623
