| L(s) = 1 | + 2.41·2-s + 3.82·4-s − 5-s − 2.78·7-s + 4.40·8-s − 2.41·10-s − 5.33·11-s + 4.82·13-s − 6.71·14-s + 2.98·16-s + 4.59·17-s − 4.21·19-s − 3.82·20-s − 12.8·22-s − 7.27·23-s + 25-s + 11.6·26-s − 10.6·28-s − 4.59·29-s + 3.02·31-s − 1.60·32-s + 11.0·34-s + 2.78·35-s − 7.59·37-s − 10.1·38-s − 4.40·40-s − 9.91·41-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s − 0.447·5-s − 1.05·7-s + 1.55·8-s − 0.763·10-s − 1.60·11-s + 1.33·13-s − 1.79·14-s + 0.746·16-s + 1.11·17-s − 0.966·19-s − 0.855·20-s − 2.74·22-s − 1.51·23-s + 0.200·25-s + 2.28·26-s − 2.01·28-s − 0.852·29-s + 0.543·31-s − 0.284·32-s + 1.90·34-s + 0.470·35-s − 1.24·37-s − 1.65·38-s − 0.696·40-s − 1.54·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 7 | \( 1 + 2.78T + 7T^{2} \) |
| 11 | \( 1 + 5.33T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 - 4.59T + 17T^{2} \) |
| 19 | \( 1 + 4.21T + 19T^{2} \) |
| 23 | \( 1 + 7.27T + 23T^{2} \) |
| 29 | \( 1 + 4.59T + 29T^{2} \) |
| 31 | \( 1 - 3.02T + 31T^{2} \) |
| 37 | \( 1 + 7.59T + 37T^{2} \) |
| 41 | \( 1 + 9.91T + 41T^{2} \) |
| 43 | \( 1 + 9.15T + 43T^{2} \) |
| 47 | \( 1 - 0.816T + 47T^{2} \) |
| 53 | \( 1 - 1.22T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 5.25T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 5.89T + 71T^{2} \) |
| 73 | \( 1 - 5.57T + 73T^{2} \) |
| 79 | \( 1 + 5.27T + 79T^{2} \) |
| 83 | \( 1 + 7.03T + 83T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−8.090087795963575339231172813534, −6.91928561477970928007103830894, −6.49758324425863860514513063649, −5.54895789287169532734149542588, −5.28973057259551620859666909818, −4.02488624604754688686482360711, −3.60851082513385172229747396373, −2.91827566965156660351698877549, −1.91980766159156622423578605008, 0,
1.91980766159156622423578605008, 2.91827566965156660351698877549, 3.60851082513385172229747396373, 4.02488624604754688686482360711, 5.28973057259551620859666909818, 5.54895789287169532734149542588, 6.49758324425863860514513063649, 6.91928561477970928007103830894, 8.090087795963575339231172813534
