| L(s) = 1 | − 2-s − 3-s + 4-s − 1.89·5-s + 6-s + 1.64·7-s − 8-s + 9-s + 1.89·10-s − 3.44·11-s − 12-s − 5.40·13-s − 1.64·14-s + 1.89·15-s + 16-s − 17-s − 18-s − 5.05·19-s − 1.89·20-s − 1.64·21-s + 3.44·22-s − 7.56·23-s + 24-s − 1.40·25-s + 5.40·26-s − 27-s + 1.64·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.848·5-s + 0.408·6-s + 0.620·7-s − 0.353·8-s + 0.333·9-s + 0.599·10-s − 1.03·11-s − 0.288·12-s − 1.49·13-s − 0.438·14-s + 0.489·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 1.16·19-s − 0.424·20-s − 0.358·21-s + 0.734·22-s − 1.57·23-s + 0.204·24-s − 0.280·25-s + 1.06·26-s − 0.192·27-s + 0.310·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1389092007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1389092007\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 + 1.89T + 5T^{2} \) |
| 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 + 3.44T + 11T^{2} \) |
| 13 | \( 1 + 5.40T + 13T^{2} \) |
| 19 | \( 1 + 5.05T + 19T^{2} \) |
| 23 | \( 1 + 7.56T + 23T^{2} \) |
| 29 | \( 1 + 2.54T + 29T^{2} \) |
| 31 | \( 1 - 1.76T + 31T^{2} \) |
| 37 | \( 1 + 2.74T + 37T^{2} \) |
| 41 | \( 1 + 1.58T + 41T^{2} \) |
| 43 | \( 1 - 1.82T + 43T^{2} \) |
| 47 | \( 1 + 7.14T + 47T^{2} \) |
| 53 | \( 1 + 3.26T + 53T^{2} \) |
| 61 | \( 1 - 3.88T + 61T^{2} \) |
| 67 | \( 1 + 3.05T + 67T^{2} \) |
| 71 | \( 1 + 1.57T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 5.95T + 79T^{2} \) |
| 83 | \( 1 - 9.77T + 83T^{2} \) |
| 89 | \( 1 + 5.77T + 89T^{2} \) |
| 97 | \( 1 - 8.85T + 97T^{2} \) |
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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
−7.948950601096013329296800026297, −7.62895309853754437841493837025, −6.82678127083516889912836341248, −6.04286575931930955283048875476, −5.13394825220520870748541703409, −4.57874783328132298723255834033, −3.71606118523923375190290613333, −2.47920776415777301988879461710, −1.83621545593640256717639528314, −0.21141054323432542135705877913,
0.21141054323432542135705877913, 1.83621545593640256717639528314, 2.47920776415777301988879461710, 3.71606118523923375190290613333, 4.57874783328132298723255834033, 5.13394825220520870748541703409, 6.04286575931930955283048875476, 6.82678127083516889912836341248, 7.62895309853754437841493837025, 7.948950601096013329296800026297
