| L(s) = 1 | − 2-s + 3-s + 4-s − 3.19·5-s − 6-s + 2.75·7-s − 8-s + 9-s + 3.19·10-s − 5.58·11-s + 12-s − 0.357·13-s − 2.75·14-s − 3.19·15-s + 16-s + 6.64·17-s − 18-s + 19-s − 3.19·20-s + 2.75·21-s + 5.58·22-s − 8.90·23-s − 24-s + 5.21·25-s + 0.357·26-s + 27-s + 2.75·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.42·5-s − 0.408·6-s + 1.04·7-s − 0.353·8-s + 0.333·9-s + 1.01·10-s − 1.68·11-s + 0.288·12-s − 0.0992·13-s − 0.736·14-s − 0.825·15-s + 0.250·16-s + 1.61·17-s − 0.235·18-s + 0.229·19-s − 0.714·20-s + 0.601·21-s + 1.19·22-s − 1.85·23-s − 0.204·24-s + 1.04·25-s + 0.0702·26-s + 0.192·27-s + 0.521·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.049096700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.049096700\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 + 3.19T + 5T^{2} \) |
| 7 | \( 1 - 2.75T + 7T^{2} \) |
| 11 | \( 1 + 5.58T + 11T^{2} \) |
| 13 | \( 1 + 0.357T + 13T^{2} \) |
| 17 | \( 1 - 6.64T + 17T^{2} \) |
| 23 | \( 1 + 8.90T + 23T^{2} \) |
| 29 | \( 1 + 8.60T + 29T^{2} \) |
| 31 | \( 1 + 9.83T + 31T^{2} \) |
| 37 | \( 1 - 9.10T + 37T^{2} \) |
| 41 | \( 1 - 6.09T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 59 | \( 1 - 9.77T + 59T^{2} \) |
| 61 | \( 1 - 6.88T + 61T^{2} \) |
| 67 | \( 1 + 2.61T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 7.87T + 73T^{2} \) |
| 79 | \( 1 + 2.48T + 79T^{2} \) |
| 83 | \( 1 - 1.18T + 83T^{2} \) |
| 89 | \( 1 - 3.43T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−7.972512446808000390936981423115, −7.55175130354245931282208456091, −7.41576633946728638343177375778, −5.77997948826482949900277791976, −5.32545916178072124101235120061, −4.18875169778734650301359825309, −3.66342249041775504811960906643, −2.66653434985535185331308926383, −1.85223516200264583370062867159, −0.56884334032789927607090789390,
0.56884334032789927607090789390, 1.85223516200264583370062867159, 2.66653434985535185331308926383, 3.66342249041775504811960906643, 4.18875169778734650301359825309, 5.32545916178072124101235120061, 5.77997948826482949900277791976, 7.41576633946728638343177375778, 7.55175130354245931282208456091, 7.972512446808000390936981423115
