L(s) = 1 | + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (−0.707 − 1.70i)11-s + (−1 − i)23-s + (0.707 − 0.707i)25-s + (0.707 − 1.70i)29-s + (−0.707 + 0.292i)37-s + (0.292 + 0.707i)43-s − 1.00i·49-s + (0.292 + 0.707i)53-s + 1.00·63-s + (−0.292 + 0.707i)67-s + (1.70 + 0.707i)77-s − 1.41i·79-s + 1.00i·81-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (−0.707 − 1.70i)11-s + (−1 − i)23-s + (0.707 − 0.707i)25-s + (0.707 − 1.70i)29-s + (−0.707 + 0.292i)37-s + (0.292 + 0.707i)43-s − 1.00i·49-s + (0.292 + 0.707i)53-s + 1.00·63-s + (−0.292 + 0.707i)67-s + (1.70 + 0.707i)77-s − 1.41i·79-s + 1.00i·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6798158388\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6798158388\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 5 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.076558514745860027912054774735, −8.504441143507221512841261413350, −7.970029632678371860905025494392, −6.49082120131278906716560222498, −6.13333274453785129025115827209, −5.43910631144320791523523074972, −4.16867116318732167491903933302, −3.06884275205485604955753483756, −2.57831894953406648589561554022, −0.48535134911256242876957148613,
1.74648958610442668539868404514, 2.83770628128964537346488959231, 3.83404111030197121851626879394, 4.90697196190612122409324870905, 5.48419289521583337998148257676, 6.74348346996390722463628708819, 7.28633170730454329317776208569, 7.982947121376490576155660547394, 8.983005632727111375720327826975, 9.779676498422551207550154861340