L(s) = 1 | + (0.951 − 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (−0.951 + 0.309i)5-s + (0.809 + 0.587i)6-s + 1.17i·7-s + (0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (−0.587 − 1.80i)11-s + (0.951 + 0.309i)12-s + (0.363 + 1.11i)14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + i·18-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (−0.951 + 0.309i)5-s + (0.809 + 0.587i)6-s + 1.17i·7-s + (0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (−0.587 − 1.80i)11-s + (0.951 + 0.309i)12-s + (0.363 + 1.11i)14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.581001672\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.581001672\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 3 | \( 1 + (-0.587 - 0.809i)T \) |
| 5 | \( 1 + (0.951 - 0.309i)T \) |
good | 7 | \( 1 - 1.17iT - T^{2} \) |
| 11 | \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
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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
−11.11900594879586665299460782340, −10.37401849557678662909036382178, −9.128723138257246690176954062734, −8.380316672656604286168766163721, −7.46816133755269802001550143596, −5.96993959085859948239202739192, −5.35697832941271490060252648184, −4.13910739073987774678598460498, −3.24759096176251271887635550111, −2.54325087977839769821938099747,
1.83921522235390678757505326512, 3.30330636615699069099064835231, 4.19691112810515798177461400369, 5.06327270827912649708680498399, 6.64420071256476009707830287119, 7.44266877972036742124913547966, 7.58389213235734221609752691853, 8.738320322643751399583016296997, 10.07425996752919593607193674229, 11.08318840088090107780214364253