| L(s) = 1 | + (−1.38 + 0.274i)2-s + (0.652 + 1.28i)3-s + (1.84 − 0.762i)4-s + (2.03 − 0.925i)5-s + (−1.25 − 1.59i)6-s + 1.67i·7-s + (−2.35 + 1.56i)8-s + (0.549 − 0.756i)9-s + (−2.56 + 1.84i)10-s + (−0.376 − 0.0595i)11-s + (2.18 + 1.86i)12-s + (5.88 − 0.931i)13-s + (−0.461 − 2.33i)14-s + (2.51 + 2.00i)15-s + (2.83 − 2.81i)16-s + (1.49 + 4.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.980 + 0.194i)2-s + (0.376 + 0.739i)3-s + (0.924 − 0.381i)4-s + (0.910 − 0.413i)5-s + (−0.513 − 0.651i)6-s + 0.634i·7-s + (−0.832 + 0.553i)8-s + (0.183 − 0.252i)9-s + (−0.812 + 0.582i)10-s + (−0.113 − 0.0179i)11-s + (0.629 + 0.539i)12-s + (1.63 − 0.258i)13-s + (−0.123 − 0.622i)14-s + (0.648 + 0.517i)15-s + (0.709 − 0.704i)16-s + (0.363 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16449 + 0.451830i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16449 + 0.451830i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 - 0.274i)T \) |
| 5 | \( 1 + (-2.03 + 0.925i)T \) |
| good | 3 | \( 1 + (-0.652 - 1.28i)T + (-1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 - 1.67iT - 7T^{2} \) |
| 11 | \( 1 + (0.376 + 0.0595i)T + (10.4 + 3.39i)T^{2} \) |
| 13 | \( 1 + (-5.88 + 0.931i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.49 - 4.61i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (5.97 + 3.04i)T + (11.1 + 15.3i)T^{2} \) |
| 23 | \( 1 + (4.03 + 5.55i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.964 - 1.89i)T + (-17.0 + 23.4i)T^{2} \) |
| 31 | \( 1 + (-1.74 - 5.36i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.233 - 1.47i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (0.668 - 0.920i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-4.99 - 4.99i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.93 - 5.95i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (12.4 - 6.36i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (1.85 + 11.7i)T + (-56.1 + 18.2i)T^{2} \) |
| 61 | \( 1 + (1.35 - 8.53i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (1.76 + 0.899i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (8.22 + 2.67i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (6.36 + 8.75i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.52 + 7.76i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.69 - 3.40i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (5.91 + 8.14i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.60 + 11.1i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.70651471679535301377075035379, −10.50207503574615520290866747567, −9.373875476812349957525825299505, −8.720296386038211759479508047185, −8.255783602259327160341248889436, −6.35988419936744665865540640586, −6.07041019102626814873341749570, −4.52180465791094065793349586652, −2.96268628713763779274132109056, −1.51982425756154666603351131522,
1.37022151740766313829222294787, 2.36024884104081847245133977344, 3.79635018230778072435922817940, 5.86989491681701499254861739218, 6.68030887576964758598544437867, 7.56676996921378397398129370441, 8.344877211088822112885022366537, 9.375966916207770249233176871493, 10.26722666602974077459395316353, 10.88486375731419341543183710536
