L(s) = 1 | + 1.87·2-s − 3-s + 1.52·4-s + 5-s − 1.87·6-s + 4.96·7-s − 0.900·8-s + 9-s + 1.87·10-s + 4.54·11-s − 1.52·12-s + 0.131·13-s + 9.31·14-s − 15-s − 4.72·16-s + 4.41·17-s + 1.87·18-s − 0.535·19-s + 1.52·20-s − 4.96·21-s + 8.51·22-s − 6.91·23-s + 0.900·24-s + 25-s + 0.247·26-s − 27-s + 7.54·28-s + ⋯ |
L(s) = 1 | + 1.32·2-s − 0.577·3-s + 0.760·4-s + 0.447·5-s − 0.765·6-s + 1.87·7-s − 0.318·8-s + 0.333·9-s + 0.593·10-s + 1.36·11-s − 0.438·12-s + 0.0365·13-s + 2.48·14-s − 0.258·15-s − 1.18·16-s + 1.07·17-s + 0.442·18-s − 0.122·19-s + 0.339·20-s − 1.08·21-s + 1.81·22-s − 1.44·23-s + 0.183·24-s + 0.200·25-s + 0.0485·26-s − 0.192·27-s + 1.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.079888679\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.079888679\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 - 1.87T + 2T^{2} \) |
| 7 | \( 1 - 4.96T + 7T^{2} \) |
| 11 | \( 1 - 4.54T + 11T^{2} \) |
| 13 | \( 1 - 0.131T + 13T^{2} \) |
| 17 | \( 1 - 4.41T + 17T^{2} \) |
| 19 | \( 1 + 0.535T + 19T^{2} \) |
| 23 | \( 1 + 6.91T + 23T^{2} \) |
| 29 | \( 1 - 9.49T + 29T^{2} \) |
| 31 | \( 1 - 6.27T + 31T^{2} \) |
| 37 | \( 1 + 0.641T + 37T^{2} \) |
| 41 | \( 1 - 4.96T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 + 6.53T + 47T^{2} \) |
| 53 | \( 1 - 5.80T + 53T^{2} \) |
| 59 | \( 1 - 3.71T + 59T^{2} \) |
| 61 | \( 1 - 7.38T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 3.25T + 71T^{2} \) |
| 73 | \( 1 - 2.14T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.249455301484387944410391496159, −7.00614405410183063538782366665, −6.43544412032849151240838758847, −5.69515679723967001505601902626, −5.20625750028439905849656258094, −4.39352053367872282939886372095, −4.12279834153426580515021343907, −2.94975873038742264065657274401, −1.84984387774097295363245640558, −1.12047032451496325086027813291,
1.12047032451496325086027813291, 1.84984387774097295363245640558, 2.94975873038742264065657274401, 4.12279834153426580515021343907, 4.39352053367872282939886372095, 5.20625750028439905849656258094, 5.69515679723967001505601902626, 6.43544412032849151240838758847, 7.00614405410183063538782366665, 8.249455301484387944410391496159