L(s) = 1 | + i·2-s + (−2.18 − 2.18i)3-s − 4-s + (1.71 − 1.43i)5-s + (2.18 − 2.18i)6-s + (−3.43 − 3.43i)7-s − i·8-s + 6.54i·9-s + (1.43 + 1.71i)10-s + 3.68i·11-s + (2.18 + 2.18i)12-s + 1.41i·13-s + (3.43 − 3.43i)14-s + (−6.87 − 0.629i)15-s + 16-s − 4.10·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−1.26 − 1.26i)3-s − 0.5·4-s + (0.768 − 0.639i)5-s + (0.891 − 0.891i)6-s + (−1.29 − 1.29i)7-s − 0.353i·8-s + 2.18i·9-s + (0.452 + 0.543i)10-s + 1.10i·11-s + (0.630 + 0.630i)12-s + 0.391i·13-s + (0.918 − 0.918i)14-s + (−1.77 − 0.162i)15-s + 0.250·16-s − 0.996·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00806507 + 0.208622i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00806507 + 0.208622i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (-1.71 + 1.43i)T \) |
| 37 | \( 1 + (3.54 - 4.94i)T \) |
good | 3 | \( 1 + (2.18 + 2.18i)T + 3iT^{2} \) |
| 7 | \( 1 + (3.43 + 3.43i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.68iT - 11T^{2} \) |
| 13 | \( 1 - 1.41iT - 13T^{2} \) |
| 17 | \( 1 + 4.10T + 17T^{2} \) |
| 19 | \( 1 + (1.16 + 1.16i)T + 19iT^{2} \) |
| 23 | \( 1 - 5.46iT - 23T^{2} \) |
| 29 | \( 1 + (1.45 - 1.45i)T - 29iT^{2} \) |
| 31 | \( 1 + (6.49 + 6.49i)T + 31iT^{2} \) |
| 41 | \( 1 + 3.44iT - 41T^{2} \) |
| 43 | \( 1 + 6.72iT - 43T^{2} \) |
| 47 | \( 1 + (-1.97 - 1.97i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.68 + 2.68i)T - 53iT^{2} \) |
| 59 | \( 1 + (3.35 + 3.35i)T + 59iT^{2} \) |
| 61 | \( 1 + (5.14 + 5.14i)T + 61iT^{2} \) |
| 67 | \( 1 + (4.59 - 4.59i)T - 67iT^{2} \) |
| 71 | \( 1 + 1.17T + 71T^{2} \) |
| 73 | \( 1 + (2.75 + 2.75i)T + 73iT^{2} \) |
| 79 | \( 1 + (-5.63 - 5.63i)T + 79iT^{2} \) |
| 83 | \( 1 + (1.45 - 1.45i)T - 83iT^{2} \) |
| 89 | \( 1 + (-7.40 + 7.40i)T - 89iT^{2} \) |
| 97 | \( 1 + 4.36T + 97T^{2} \) |
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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
−10.92356124010965415144336802091, −9.983636518433535642938832429928, −9.137590455179072071641868713300, −7.53092429994376530384620945435, −6.95343909341544859605534506415, −6.34053724633025663923350085473, −5.36882671634449460895952656153, −4.24979060406134618691803002388, −1.75175067077963828433399166578, −0.15802028102805736043198459690,
2.70339034305590081894579535867, 3.66294778312106569969647965466, 5.15911021573426063951632492042, 5.96922821586205881327721604406, 6.46517526696266272816702406405, 8.941248762008040727816407532645, 9.259370764197386837534485431513, 10.42270901820036058648760661377, 10.67744142844087493169833158608, 11.64852466275371513107935977824