L(s) = 1 | − 2i·3-s + i·7-s − 9-s − 2i·17-s + 2·19-s + 2·21-s − 8i·23-s − 4i·27-s − 2·29-s + 4·31-s − 6i·37-s − 2·41-s − 8i·43-s − 4i·47-s − 49-s + ⋯ |
L(s) = 1 | − 1.15i·3-s + 0.377i·7-s − 0.333·9-s − 0.485i·17-s + 0.458·19-s + 0.436·21-s − 1.66i·23-s − 0.769i·27-s − 0.371·29-s + 0.718·31-s − 0.986i·37-s − 0.312·41-s − 1.21i·43-s − 0.583i·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.508546354\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.508546354\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.144724936973781732491592415782, −8.397889481519338765926418803948, −7.60097980278152930322486802212, −6.89597773728039066374595571624, −6.19706853742635374274902779089, −5.25351275791154087860041031210, −4.18472634125489876184966099370, −2.82993948735686424815583918179, −1.96510182166407410189717775949, −0.64455666675166078320055723254,
1.45497257569858995093002737719, 3.10451445644428243907641366325, 3.84425622099164970912232649079, 4.71208934799557050061407574633, 5.46989344098252350302431261500, 6.50027241114960682451324655593, 7.49480057458445876057315126169, 8.267844583602414184539988046573, 9.325804699386767276234996236196, 9.782137192299039015420428127973