| L(s) = 1 | + i·3-s − 4.41i·7-s − 9-s + 4.45·11-s − 5.74i·13-s − 6.57i·17-s + 2.78·19-s + 4.41·21-s + 1.31i·23-s − i·27-s + 4.84·29-s − 0.197·31-s + 4.45i·33-s + 8.20i·37-s + 5.74·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.66i·7-s − 0.333·9-s + 1.34·11-s − 1.59i·13-s − 1.59i·17-s + 0.638·19-s + 0.963·21-s + 0.274i·23-s − 0.192i·27-s + 0.899·29-s − 0.0354·31-s + 0.774i·33-s + 1.34i·37-s + 0.920·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.179014037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.179014037\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.41iT - 7T^{2} \) |
| 11 | \( 1 - 4.45T + 11T^{2} \) |
| 13 | \( 1 + 5.74iT - 13T^{2} \) |
| 17 | \( 1 + 6.57iT - 17T^{2} \) |
| 19 | \( 1 - 2.78T + 19T^{2} \) |
| 23 | \( 1 - 1.31iT - 23T^{2} \) |
| 29 | \( 1 - 4.84T + 29T^{2} \) |
| 31 | \( 1 + 0.197T + 31T^{2} \) |
| 37 | \( 1 - 8.20iT - 37T^{2} \) |
| 41 | \( 1 - 7.84T + 41T^{2} \) |
| 43 | \( 1 + 0.412iT - 43T^{2} \) |
| 47 | \( 1 + 7.79iT - 47T^{2} \) |
| 53 | \( 1 + 0.315iT - 53T^{2} \) |
| 59 | \( 1 + 2.51T + 59T^{2} \) |
| 61 | \( 1 - 9.33T + 61T^{2} \) |
| 67 | \( 1 + 11.9iT - 67T^{2} \) |
| 71 | \( 1 - 0.509T + 71T^{2} \) |
| 73 | \( 1 - 15.7iT - 73T^{2} \) |
| 79 | \( 1 - 3.43T + 79T^{2} \) |
| 83 | \( 1 + 5.37iT - 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 5.07iT - 97T^{2} \) |
| 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
−7.66619054044813278778710735731, −7.04444298218201591237217790282, −6.47335620327058982759578549236, −5.44378184873776226879145521129, −4.83171994145500554746664695191, −4.08102418365794270171843849918, −3.43755123595865894534030917700, −2.78491476234317016563611256704, −1.16085590518996643238373577967, −0.60317777121660434109388601838,
1.28120109290345947555498479493, 1.96296770666748212887740398629, 2.69058088172431249498285379385, 3.80424363423664046159824226449, 4.42273107837061777430727081556, 5.48353482784072331097534235197, 6.17485727370306434857624247315, 6.45380373180864495689473575224, 7.28718629000862827356327106355, 8.195519975549805180979239041880
