| L(s) = 1 | − 1.80·2-s − 3-s + 1.25·4-s − 3.82·5-s + 1.80·6-s + 7-s + 1.35·8-s + 9-s + 6.89·10-s − 1.43·11-s − 1.25·12-s + 6.02·13-s − 1.80·14-s + 3.82·15-s − 4.93·16-s + 3.50·17-s − 1.80·18-s + 2.91·19-s − 4.78·20-s − 21-s + 2.58·22-s + 0.715·23-s − 1.35·24-s + 9.61·25-s − 10.8·26-s − 27-s + 1.25·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s − 0.577·3-s + 0.625·4-s − 1.70·5-s + 0.736·6-s + 0.377·7-s + 0.477·8-s + 0.333·9-s + 2.17·10-s − 0.432·11-s − 0.361·12-s + 1.67·13-s − 0.481·14-s + 0.987·15-s − 1.23·16-s + 0.850·17-s − 0.424·18-s + 0.669·19-s − 1.06·20-s − 0.218·21-s + 0.551·22-s + 0.149·23-s − 0.275·24-s + 1.92·25-s − 2.13·26-s − 0.192·27-s + 0.236·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6202728325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6202728325\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 5 | \( 1 + 3.82T + 5T^{2} \) |
| 11 | \( 1 + 1.43T + 11T^{2} \) |
| 13 | \( 1 - 6.02T + 13T^{2} \) |
| 17 | \( 1 - 3.50T + 17T^{2} \) |
| 19 | \( 1 - 2.91T + 19T^{2} \) |
| 23 | \( 1 - 0.715T + 23T^{2} \) |
| 29 | \( 1 - 7.30T + 29T^{2} \) |
| 31 | \( 1 - 2.00T + 31T^{2} \) |
| 37 | \( 1 - 5.89T + 37T^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 43 | \( 1 - 1.13T + 43T^{2} \) |
| 47 | \( 1 + 2.43T + 47T^{2} \) |
| 53 | \( 1 - 6.26T + 53T^{2} \) |
| 59 | \( 1 + 9.03T + 59T^{2} \) |
| 61 | \( 1 - 8.11T + 61T^{2} \) |
| 67 | \( 1 - 4.55T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 - 9.24T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 + 6.95T + 83T^{2} \) |
| 89 | \( 1 + 1.56T + 89T^{2} \) |
| 97 | \( 1 - 9.54T + 97T^{2} \) |
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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
−8.236640685457920256584023981210, −8.001344544973428059041086468393, −7.28802406686525365661818958299, −6.55065315655788392738046941419, −5.49670639807330734952243374511, −4.55808265419811591138717995684, −3.93470570167605861054467968304, −2.95460912123456538733346686549, −1.26139323166470584150324807356, −0.67961045843755055039550110536,
0.67961045843755055039550110536, 1.26139323166470584150324807356, 2.95460912123456538733346686549, 3.93470570167605861054467968304, 4.55808265419811591138717995684, 5.49670639807330734952243374511, 6.55065315655788392738046941419, 7.28802406686525365661818958299, 8.001344544973428059041086468393, 8.236640685457920256584023981210
