L(s) = 1 | + (−1.40 − 0.146i)2-s + (0.169 + 1.72i)3-s + (1.95 + 0.413i)4-s − 2.76i·5-s + (0.0141 − 2.44i)6-s − 3.82i·7-s + (−2.69 − 0.868i)8-s + (−2.94 + 0.585i)9-s + (−0.406 + 3.89i)10-s − 2.92·11-s + (−0.379 + 3.44i)12-s − 4.70·13-s + (−0.561 + 5.38i)14-s + (4.76 − 0.469i)15-s + (3.65 + 1.61i)16-s − 4.06i·17-s + ⋯ |
L(s) = 1 | + (−0.994 − 0.103i)2-s + (0.0980 + 0.995i)3-s + (0.978 + 0.206i)4-s − 1.23i·5-s + (0.00579 − 0.999i)6-s − 1.44i·7-s + (−0.951 − 0.307i)8-s + (−0.980 + 0.195i)9-s + (−0.128 + 1.23i)10-s − 0.883·11-s + (−0.109 + 0.993i)12-s − 1.30·13-s + (−0.150 + 1.43i)14-s + (1.23 − 0.121i)15-s + (0.914 + 0.404i)16-s − 0.985i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.380112 - 0.424327i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.380112 - 0.424327i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 + 0.146i)T \) |
| 3 | \( 1 + (-0.169 - 1.72i)T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 2.76iT - 5T^{2} \) |
| 7 | \( 1 + 3.82iT - 7T^{2} \) |
| 11 | \( 1 + 2.92T + 11T^{2} \) |
| 13 | \( 1 + 4.70T + 13T^{2} \) |
| 17 | \( 1 + 4.06iT - 17T^{2} \) |
| 19 | \( 1 + 3.75iT - 19T^{2} \) |
| 29 | \( 1 - 8.01iT - 29T^{2} \) |
| 31 | \( 1 + 1.15iT - 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 1.54iT - 41T^{2} \) |
| 43 | \( 1 + 2.22iT - 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 6.30iT - 53T^{2} \) |
| 59 | \( 1 + 7.03T + 59T^{2} \) |
| 61 | \( 1 + 6.15T + 61T^{2} \) |
| 67 | \( 1 - 1.65iT - 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 5.68T + 73T^{2} \) |
| 79 | \( 1 - 3.06iT - 79T^{2} \) |
| 83 | \( 1 + 2.88T + 83T^{2} \) |
| 89 | \( 1 + 8.17iT - 89T^{2} \) |
| 97 | \( 1 + 4.28T + 97T^{2} \) |
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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
−11.24712680038896876840728260326, −10.49911024221925137132468376245, −9.654840461087711188864296333123, −9.035285222146517074832551548123, −7.905352335645394724190708940460, −7.12016626763835983709706284179, −5.26428810609676874017099043362, −4.43786046395680433575998954721, −2.81248625928597126450061153501, −0.55057673232534083751043900377,
2.25792569689683547052017900177, 2.77794448176161781270699095796, 5.71475185023637923914633719201, 6.31866829099666055748011104551, 7.53156843267487049040046113421, 8.030263495155313903085055538212, 9.203041898647918150314702256618, 10.22090282496838923037866739387, 11.15283494318419760497985777146, 12.07705317860618361565907451476