L(s) = 1 | − 2.27·2-s + 3.18·4-s + 2.98·5-s − 2.44·7-s − 2.70·8-s − 6.79·10-s − 3.22·11-s − 2.73·13-s + 5.57·14-s − 0.220·16-s − 2.14·17-s + 1.46·19-s + 9.50·20-s + 7.34·22-s + 23-s + 3.90·25-s + 6.22·26-s − 7.79·28-s − 29-s + 6.98·31-s + 5.90·32-s + 4.87·34-s − 7.30·35-s + 10.5·37-s − 3.34·38-s − 8.06·40-s − 6.90·41-s + ⋯ |
L(s) = 1 | − 1.61·2-s + 1.59·4-s + 1.33·5-s − 0.924·7-s − 0.955·8-s − 2.14·10-s − 0.972·11-s − 0.757·13-s + 1.48·14-s − 0.0551·16-s − 0.519·17-s + 0.336·19-s + 2.12·20-s + 1.56·22-s + 0.208·23-s + 0.781·25-s + 1.21·26-s − 1.47·28-s − 0.185·29-s + 1.25·31-s + 1.04·32-s + 0.836·34-s − 1.23·35-s + 1.73·37-s − 0.542·38-s − 1.27·40-s − 1.07·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6812803205\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6812803205\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 5 | \( 1 - 2.98T + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 3.22T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 + 2.14T + 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 31 | \( 1 - 6.98T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 6.90T + 41T^{2} \) |
| 43 | \( 1 + 9.50T + 43T^{2} \) |
| 47 | \( 1 + 3.28T + 47T^{2} \) |
| 53 | \( 1 + 7.20T + 53T^{2} \) |
| 59 | \( 1 - 7.50T + 59T^{2} \) |
| 61 | \( 1 + 0.996T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 5.24T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 + 7.48T + 79T^{2} \) |
| 83 | \( 1 + 7.33T + 83T^{2} \) |
| 89 | \( 1 - 7.50T + 89T^{2} \) |
| 97 | \( 1 - 8.72T + 97T^{2} \) |
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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
−8.195700607854632245806457483260, −7.54436324534547327313795538655, −6.64711351985719013837018825143, −6.37614042224701054716548606485, −5.38996134557264362500564966212, −4.65397874201562907778759483880, −3.10139630666940662340778738087, −2.47443877579885311215072125644, −1.74714351386224145405821349410, −0.53735737997428305345268284130,
0.53735737997428305345268284130, 1.74714351386224145405821349410, 2.47443877579885311215072125644, 3.10139630666940662340778738087, 4.65397874201562907778759483880, 5.38996134557264362500564966212, 6.37614042224701054716548606485, 6.64711351985719013837018825143, 7.54436324534547327313795538655, 8.195700607854632245806457483260