| 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 | \( 1 + 5.07iT - 97T^{2} \) |
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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
−8.195519975549805180979239041880, −7.28718629000862827356327106355, −6.45380373180864495689473575224, −6.17485727370306434857624247315, −5.48353482784072331097534235197, −4.42273107837061777430727081556, −3.80424363423664046159824226449, −2.69058088172431249498285379385, −1.96296770666748212887740398629, −1.28120109290345947555498479493,
0.60317777121660434109388601838, 1.16085590518996643238373577967, 2.78491476234317016563611256704, 3.43755123595865894534030917700, 4.08102418365794270171843849918, 4.83171994145500554746664695191, 5.44378184873776226879145521129, 6.47335620327058982759578549236, 7.04444298218201591237217790282, 7.66619054044813278778710735731
