L(s) = 1 | + i·2-s + (0.707 − 0.707i)3-s − 4-s + (−2.17 − 0.530i)5-s + (0.707 + 0.707i)6-s + (−1.93 + 1.93i)7-s − i·8-s − 1.00i·9-s + (0.530 − 2.17i)10-s + 2.54i·11-s + (−0.707 + 0.707i)12-s − 2.64i·13-s + (−1.93 − 1.93i)14-s + (−1.91 + 1.16i)15-s + 16-s + 3.60·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.408 − 0.408i)3-s − 0.5·4-s + (−0.971 − 0.237i)5-s + (0.288 + 0.288i)6-s + (−0.730 + 0.730i)7-s − 0.353i·8-s − 0.333i·9-s + (0.167 − 0.686i)10-s + 0.766i·11-s + (−0.204 + 0.204i)12-s − 0.732i·13-s + (−0.516 − 0.516i)14-s + (−0.493 + 0.299i)15-s + 0.250·16-s + 0.874·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.104483641\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.104483641\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (2.17 + 0.530i)T \) |
| 37 | \( 1 + (1.59 + 5.87i)T \) |
good | 7 | \( 1 + (1.93 - 1.93i)T - 7iT^{2} \) |
| 11 | \( 1 - 2.54iT - 11T^{2} \) |
| 13 | \( 1 + 2.64iT - 13T^{2} \) |
| 17 | \( 1 - 3.60T + 17T^{2} \) |
| 19 | \( 1 + (-2.47 + 2.47i)T - 19iT^{2} \) |
| 23 | \( 1 + 4.49iT - 23T^{2} \) |
| 29 | \( 1 + (4.58 + 4.58i)T + 29iT^{2} \) |
| 31 | \( 1 + (-7.33 + 7.33i)T - 31iT^{2} \) |
| 41 | \( 1 - 4.31iT - 41T^{2} \) |
| 43 | \( 1 + 3.12iT - 43T^{2} \) |
| 47 | \( 1 + (-4.51 + 4.51i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.301 + 0.301i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.813 + 0.813i)T - 59iT^{2} \) |
| 61 | \( 1 + (0.867 - 0.867i)T - 61iT^{2} \) |
| 67 | \( 1 + (5.77 + 5.77i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.30T + 71T^{2} \) |
| 73 | \( 1 + (-3.70 + 3.70i)T - 73iT^{2} \) |
| 79 | \( 1 + (-4.70 + 4.70i)T - 79iT^{2} \) |
| 83 | \( 1 + (0.524 + 0.524i)T + 83iT^{2} \) |
| 89 | \( 1 + (6.69 + 6.69i)T + 89iT^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−9.489115672918530171823384375856, −8.830182880942106359988980896896, −7.86325516762031301902483320153, −7.52230452789889123064474487771, −6.49425183645858055101039768673, −5.63879085672313021118532787585, −4.59415885416607206716140180586, −3.55536423532633559659551255829, −2.53868523576464610137771650553, −0.54374969018397289561139584174,
1.19937697826369601965326326052, 3.09623396273592612128167504741, 3.47373020113267931554773661349, 4.32040788602255958587218236356, 5.45390439796690000410787320005, 6.74754798328181428572013624163, 7.58082388289025487110195537758, 8.391591524230820349118448652396, 9.230212666364959446719903423494, 10.05463448889283687510763724006