| L(s) = 1 | − 3-s − 5-s + 2.44·7-s + 9-s − 4.44·13-s + 15-s + 2·17-s − 4.89·19-s − 2.44·21-s − 2.89·23-s + 25-s − 27-s + 0.449·29-s + 31-s − 2.44·35-s − 3.55·37-s + 4.44·39-s + 7.79·41-s − 3.10·43-s − 45-s + 8.89·47-s − 1.00·49-s − 2·51-s − 6·53-s + 4.89·57-s − 7.34·59-s − 6.89·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.925·7-s + 0.333·9-s − 1.23·13-s + 0.258·15-s + 0.485·17-s − 1.12·19-s − 0.534·21-s − 0.604·23-s + 0.200·25-s − 0.192·27-s + 0.0834·29-s + 0.179·31-s − 0.414·35-s − 0.583·37-s + 0.712·39-s + 1.21·41-s − 0.472·43-s − 0.149·45-s + 1.29·47-s − 0.142·49-s − 0.280·51-s − 0.824·53-s + 0.648·57-s − 0.956·59-s − 0.883·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4.44T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4.89T + 19T^{2} \) |
| 23 | \( 1 + 2.89T + 23T^{2} \) |
| 29 | \( 1 - 0.449T + 29T^{2} \) |
| 37 | \( 1 + 3.55T + 37T^{2} \) |
| 41 | \( 1 - 7.79T + 41T^{2} \) |
| 43 | \( 1 + 3.10T + 43T^{2} \) |
| 47 | \( 1 - 8.89T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 7.34T + 59T^{2} \) |
| 61 | \( 1 + 6.89T + 61T^{2} \) |
| 67 | \( 1 + 6.44T + 67T^{2} \) |
| 71 | \( 1 - 1.55T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 2.89T + 79T^{2} \) |
| 83 | \( 1 + 1.10T + 83T^{2} \) |
| 89 | \( 1 + 4.44T + 89T^{2} \) |
| 97 | \( 1 + 5.10T + 97T^{2} \) |
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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
−8.782056008098806155403105537388, −7.891029665001142605898083553335, −7.42164356702284395664005869848, −6.44085605214833663389355966338, −5.53816424808173309014809663781, −4.69671510872681980098394481413, −4.12512090454874214751054433819, −2.71956888659405977458816066438, −1.56624231640967028021933132310, 0,
1.56624231640967028021933132310, 2.71956888659405977458816066438, 4.12512090454874214751054433819, 4.69671510872681980098394481413, 5.53816424808173309014809663781, 6.44085605214833663389355966338, 7.42164356702284395664005869848, 7.891029665001142605898083553335, 8.782056008098806155403105537388
