L(s) = 1 | − 0.930·2-s + 2.53·3-s − 1.13·4-s − 3.38·5-s − 2.35·6-s − 3.38·7-s + 2.91·8-s + 3.41·9-s + 3.14·10-s + 11-s − 2.87·12-s + 3.14·14-s − 8.57·15-s − 0.441·16-s − 7.19·17-s − 3.17·18-s − 3.06·19-s + 3.84·20-s − 8.56·21-s − 0.930·22-s + 5.94·23-s + 7.38·24-s + 6.47·25-s + 1.05·27-s + 3.83·28-s + 5.43·29-s + 7.97·30-s + ⋯ |
L(s) = 1 | − 0.657·2-s + 1.46·3-s − 0.567·4-s − 1.51·5-s − 0.961·6-s − 1.27·7-s + 1.03·8-s + 1.13·9-s + 0.996·10-s + 0.301·11-s − 0.829·12-s + 0.840·14-s − 2.21·15-s − 0.110·16-s − 1.74·17-s − 0.748·18-s − 0.703·19-s + 0.859·20-s − 1.86·21-s − 0.198·22-s + 1.24·23-s + 1.50·24-s + 1.29·25-s + 0.202·27-s + 0.725·28-s + 1.00·29-s + 1.45·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8727727082\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8727727082\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.930T + 2T^{2} \) |
| 3 | \( 1 - 2.53T + 3T^{2} \) |
| 5 | \( 1 + 3.38T + 5T^{2} \) |
| 7 | \( 1 + 3.38T + 7T^{2} \) |
| 17 | \( 1 + 7.19T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 - 5.94T + 23T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 - 9.06T + 31T^{2} \) |
| 37 | \( 1 + 4.87T + 37T^{2} \) |
| 41 | \( 1 - 0.859T + 41T^{2} \) |
| 43 | \( 1 - 5.81T + 43T^{2} \) |
| 47 | \( 1 - 8.69T + 47T^{2} \) |
| 53 | \( 1 + 9.91T + 53T^{2} \) |
| 59 | \( 1 - 7.38T + 59T^{2} \) |
| 61 | \( 1 - 5.88T + 61T^{2} \) |
| 67 | \( 1 + 1.27T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 1.40T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 2.06T + 83T^{2} \) |
| 89 | \( 1 + 0.472T + 89T^{2} \) |
| 97 | \( 1 - 6.51T + 97T^{2} \) |
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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
−8.956962881750213630007729121611, −8.611047334874522744371202295190, −8.000743124991606465768786328133, −7.11432903552842620301083730953, −6.55581720014479125449850386470, −4.67380959429910374920024909639, −4.08826682628700658419743236583, −3.35599628203444778812573493878, −2.45152119873491141180222049736, −0.63497775362486230414964094083,
0.63497775362486230414964094083, 2.45152119873491141180222049736, 3.35599628203444778812573493878, 4.08826682628700658419743236583, 4.67380959429910374920024909639, 6.55581720014479125449850386470, 7.11432903552842620301083730953, 8.000743124991606465768786328133, 8.611047334874522744371202295190, 8.956962881750213630007729121611