L(s) = 1 | − i·3-s + (2 + i)5-s − 4i·7-s − 9-s + 6·11-s + i·13-s + (1 − 2i)15-s + 4i·17-s + 2·19-s − 4·21-s + 6i·23-s + (3 + 4i)25-s + i·27-s + 10·29-s − 4·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.894 + 0.447i)5-s − 1.51i·7-s − 0.333·9-s + 1.80·11-s + 0.277i·13-s + (0.258 − 0.516i)15-s + 0.970i·17-s + 0.458·19-s − 0.872·21-s + 1.25i·23-s + (0.600 + 0.800i)25-s + 0.192i·27-s + 1.85·29-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.618500213\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.618500213\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 - i)T \) |
| 13 | \( 1 - iT \) |
good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.663198519662771624716587161983, −7.66689760879511475984574926285, −7.05381595381731855820387239897, −6.42666367018735506774205082162, −5.96078293697032060894740412650, −4.65945338962121549324356948746, −3.85582733959903566998354501762, −3.05744024629474237944457919649, −1.57864541294792856600228268624, −1.18695580078384063840761271952,
1.01865613135659887529208141765, 2.27845855051693657560688095724, 2.96158915716007055894373938844, 4.22477599879419634678394927328, 4.92785139816646120842257796339, 5.76832631751107995023179159508, 6.21263921756240562714508946679, 7.09351890909329506586324070245, 8.424829985350713828616220005511, 8.958040616746122333111201474991