L(s) = 1 | + (−1.22 + 1.22i)7-s + (−1.22 − 1.22i)13-s − i·19-s − 31-s + (−1.22 − 1.22i)43-s − 1.99i·49-s − 61-s + (1.22 − 1.22i)67-s − 2i·79-s + 2.99·91-s + (−1.22 + 1.22i)97-s − i·109-s + ⋯ |
L(s) = 1 | + (−1.22 + 1.22i)7-s + (−1.22 − 1.22i)13-s − i·19-s − 31-s + (−1.22 − 1.22i)43-s − 1.99i·49-s − 61-s + (1.22 − 1.22i)67-s − 2i·79-s + 2.99·91-s + (−1.22 + 1.22i)97-s − i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3414570620\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3414570620\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + iT - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
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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
−8.611795953374382749983518073604, −7.68707915555352933615119637245, −6.96169067137254149454552360070, −6.21486185534567738050643774781, −5.41423435235535838430389359661, −4.92237579562580777726862184890, −3.53643820291953603802348488392, −2.88096029348874040358272467948, −2.14658291800270743101589395992, −0.18709629992634672547776406917,
1.48412636655938604056864027190, 2.68553911741543721681178171093, 3.68051551916732497067708091023, 4.22831217964985383981072381300, 5.17102365277129305154966812557, 6.22094241439894432794194788668, 6.85739674258892575170937522114, 7.33445598941504071616220216361, 8.147060563538888671658238111903, 9.210487460867424970830655556009