L(s) = 1 | − 1.80·2-s + 3-s + 1.25·4-s + 5-s − 1.80·6-s + 4.01·7-s + 1.33·8-s + 9-s − 1.80·10-s − 2.16·11-s + 1.25·12-s − 5.00·13-s − 7.24·14-s + 15-s − 4.93·16-s − 1.89·17-s − 1.80·18-s + 7.80·19-s + 1.25·20-s + 4.01·21-s + 3.90·22-s + 1.33·24-s + 25-s + 9.03·26-s + 27-s + 5.05·28-s + 3.11·29-s + ⋯ |
L(s) = 1 | − 1.27·2-s + 0.577·3-s + 0.629·4-s + 0.447·5-s − 0.737·6-s + 1.51·7-s + 0.472·8-s + 0.333·9-s − 0.570·10-s − 0.651·11-s + 0.363·12-s − 1.38·13-s − 1.93·14-s + 0.258·15-s − 1.23·16-s − 0.459·17-s − 0.425·18-s + 1.79·19-s + 0.281·20-s + 0.875·21-s + 0.831·22-s + 0.272·24-s + 0.200·25-s + 1.77·26-s + 0.192·27-s + 0.954·28-s + 0.577·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.541419178\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.541419178\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 7 | \( 1 - 4.01T + 7T^{2} \) |
| 11 | \( 1 + 2.16T + 11T^{2} \) |
| 13 | \( 1 + 5.00T + 13T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 - 7.80T + 19T^{2} \) |
| 29 | \( 1 - 3.11T + 29T^{2} \) |
| 31 | \( 1 + 3.72T + 31T^{2} \) |
| 37 | \( 1 + 5.59T + 37T^{2} \) |
| 41 | \( 1 - 8.74T + 41T^{2} \) |
| 43 | \( 1 + 7.90T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 4.54T + 59T^{2} \) |
| 61 | \( 1 + 2.59T + 61T^{2} \) |
| 67 | \( 1 - 2.74T + 67T^{2} \) |
| 71 | \( 1 - 7.70T + 71T^{2} \) |
| 73 | \( 1 + 0.852T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 - 6.79T + 89T^{2} \) |
| 97 | \( 1 - 6.92T + 97T^{2} \) |
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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
−7.77515164516367327499354297488, −7.55812334788820439314350394758, −6.97635396448074519116870265337, −5.67946134245567983293930318070, −4.86093643614190617351637127375, −4.62495262424684096492946956261, −3.21598884000243264137322243701, −2.25175122119986610775823131090, −1.75611723685664540580313630964, −0.74635681495676394388303219511,
0.74635681495676394388303219511, 1.75611723685664540580313630964, 2.25175122119986610775823131090, 3.21598884000243264137322243701, 4.62495262424684096492946956261, 4.86093643614190617351637127375, 5.67946134245567983293930318070, 6.97635396448074519116870265337, 7.55812334788820439314350394758, 7.77515164516367327499354297488