L(s) = 1 | − 2-s + 4-s − 5-s + 0.487·7-s − 8-s + 10-s + 5.20·11-s + 4.12·13-s − 0.487·14-s + 16-s − 3.67·17-s + 0.430·19-s − 20-s − 5.20·22-s − 7.31·23-s + 25-s − 4.12·26-s + 0.487·28-s + 6.77·29-s + 7.80·31-s − 32-s + 3.67·34-s − 0.487·35-s − 37-s − 0.430·38-s + 40-s + 9.80·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.184·7-s − 0.353·8-s + 0.316·10-s + 1.57·11-s + 1.14·13-s − 0.130·14-s + 0.250·16-s − 0.891·17-s + 0.0986·19-s − 0.223·20-s − 1.11·22-s − 1.52·23-s + 0.200·25-s − 0.808·26-s + 0.0921·28-s + 1.25·29-s + 1.40·31-s − 0.176·32-s + 0.630·34-s − 0.0823·35-s − 0.164·37-s − 0.0697·38-s + 0.158·40-s + 1.53·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.422379129\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.422379129\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 - 0.487T + 7T^{2} \) |
| 11 | \( 1 - 5.20T + 11T^{2} \) |
| 13 | \( 1 - 4.12T + 13T^{2} \) |
| 17 | \( 1 + 3.67T + 17T^{2} \) |
| 19 | \( 1 - 0.430T + 19T^{2} \) |
| 23 | \( 1 + 7.31T + 23T^{2} \) |
| 29 | \( 1 - 6.77T + 29T^{2} \) |
| 31 | \( 1 - 7.80T + 31T^{2} \) |
| 41 | \( 1 - 9.80T + 41T^{2} \) |
| 43 | \( 1 - 4.77T + 43T^{2} \) |
| 47 | \( 1 + 3.52T + 47T^{2} \) |
| 53 | \( 1 - 7.67T + 53T^{2} \) |
| 59 | \( 1 - 0.974T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 + 1.19T + 67T^{2} \) |
| 71 | \( 1 + 8.58T + 71T^{2} \) |
| 73 | \( 1 + 7.15T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 7.16T + 97T^{2} \) |
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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
−8.602563381758484369056794133659, −8.097960159119977994936372346201, −7.20572577133695381707636689672, −6.29410643270562873963651650926, −6.10394814717993867156852841037, −4.50673064796484224749831469050, −4.02653694099066377421803355165, −2.97744751594233946518807911358, −1.76626570129276185306923217313, −0.833607448087418395372710935486,
0.833607448087418395372710935486, 1.76626570129276185306923217313, 2.97744751594233946518807911358, 4.02653694099066377421803355165, 4.50673064796484224749831469050, 6.10394814717993867156852841037, 6.29410643270562873963651650926, 7.20572577133695381707636689672, 8.097960159119977994936372346201, 8.602563381758484369056794133659