| L(s) = 1 | + (−1 + i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s + 2.44·6-s + (3.44 − 3.44i)7-s + (2 + 2i)8-s + 2.99i·9-s + 1.55·11-s + (−2.44 + 2.44i)12-s + 6.89i·14-s − 4·16-s + (−2.99 − 2.99i)18-s − 8.44·21-s + (−1.55 + 1.55i)22-s − 4.89i·24-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s − i·4-s + 0.999·6-s + (1.30 − 1.30i)7-s + (0.707 + 0.707i)8-s + 0.999i·9-s + 0.467·11-s + (−0.707 + 0.707i)12-s + 1.84i·14-s − 16-s + (−0.707 − 0.707i)18-s − 1.84·21-s + (−0.330 + 0.330i)22-s − 0.999i·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.784488 - 0.437380i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.784488 - 0.437380i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-3.44 + 3.44i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.55T + 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 + 5.34iT - 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (2.44 + 2.44i)T + 53iT^{2} \) |
| 59 | \( 1 + 15.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (11.8 + 11.8i)T + 73iT^{2} \) |
| 79 | \( 1 - 14.6iT - 79T^{2} \) |
| 83 | \( 1 + (-4 - 4i)T + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (8.79 - 8.79i)T - 97iT^{2} \) |
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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
−10.59192115481690591893328708552, −9.732550975892989242542632221219, −8.345695090598626871958649770159, −7.82665556488751458369033807556, −7.04057098920862255678748749343, −6.27071516146700591994077048588, −5.12390441831801024183499735027, −4.33095519868326383186402466556, −1.85002080543130363494218525620, −0.792171014388010220229700700432,
1.44150347672323802627405578243, 2.85433170204208892074268218745, 4.24663673862350298665058198454, 5.10209585688561658187757202988, 6.17844439028900741561326201202, 7.47226484234625763329023682085, 8.659651388033960873753009523176, 8.983837672549506152923478373205, 10.05746930216879382778843110968, 10.84842078871719229486328977163
