L(s) = 1 | + 2-s − 3.34·3-s + 4-s − 5-s − 3.34·6-s − 2.59·7-s + 8-s + 8.19·9-s − 10-s + 4.74·11-s − 3.34·12-s + 6.69·13-s − 2.59·14-s + 3.34·15-s + 16-s − 0.748·17-s + 8.19·18-s − 3.34·19-s − 20-s + 8.69·21-s + 4.74·22-s + 1.49·23-s − 3.34·24-s + 25-s + 6.69·26-s − 17.3·27-s − 2.59·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.93·3-s + 0.5·4-s − 0.447·5-s − 1.36·6-s − 0.981·7-s + 0.353·8-s + 2.73·9-s − 0.316·10-s + 1.43·11-s − 0.965·12-s + 1.85·13-s − 0.694·14-s + 0.863·15-s + 0.250·16-s − 0.181·17-s + 1.93·18-s − 0.767·19-s − 0.223·20-s + 1.89·21-s + 1.01·22-s + 0.312·23-s − 0.682·24-s + 0.200·25-s + 1.31·26-s − 3.34·27-s − 0.490·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.087519120\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.087519120\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 + 3.34T + 3T^{2} \) |
| 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 - 6.69T + 13T^{2} \) |
| 17 | \( 1 + 0.748T + 17T^{2} \) |
| 19 | \( 1 + 3.34T + 19T^{2} \) |
| 23 | \( 1 - 1.49T + 23T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 31 | \( 1 - 7.79T + 31T^{2} \) |
| 41 | \( 1 + 6.44T + 41T^{2} \) |
| 43 | \( 1 - 1.94T + 43T^{2} \) |
| 47 | \( 1 - 1.84T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 5.84T + 59T^{2} \) |
| 61 | \( 1 - 7.94T + 61T^{2} \) |
| 67 | \( 1 - 1.84T + 67T^{2} \) |
| 71 | \( 1 + 3.88T + 71T^{2} \) |
| 73 | \( 1 + 7.49T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−11.60180156604721476862487246901, −10.81050962507551854439212411753, −10.01293868207377400237488112873, −8.637363163196287707705750464790, −6.81645835521015055774884096073, −6.51753248776819812170403466852, −5.76275040668198468447716813559, −4.41345416412251328155331739465, −3.70727608921289895429051350715, −1.08962700757098009902654475464,
1.08962700757098009902654475464, 3.70727608921289895429051350715, 4.41345416412251328155331739465, 5.76275040668198468447716813559, 6.51753248776819812170403466852, 6.81645835521015055774884096073, 8.637363163196287707705750464790, 10.01293868207377400237488112873, 10.81050962507551854439212411753, 11.60180156604721476862487246901