L(s) = 1 | + i·2-s + (−1.28 + 1.28i)3-s − 4-s + (1.45 + 1.69i)5-s + (−1.28 − 1.28i)6-s + (0.579 − 0.579i)7-s − i·8-s − 0.324i·9-s + (−1.69 + 1.45i)10-s + 3.64i·11-s + (1.28 − 1.28i)12-s + 3.10i·13-s + (0.579 + 0.579i)14-s + (−4.06 − 0.301i)15-s + 16-s − 8.09·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.744 + 0.744i)3-s − 0.5·4-s + (0.652 + 0.757i)5-s + (−0.526 − 0.526i)6-s + (0.219 − 0.219i)7-s − 0.353i·8-s − 0.108i·9-s + (−0.535 + 0.461i)10-s + 1.09i·11-s + (0.372 − 0.372i)12-s + 0.861i·13-s + (0.154 + 0.154i)14-s + (−1.04 − 0.0777i)15-s + 0.250·16-s − 1.96·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0330162 + 0.942496i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0330162 + 0.942496i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.45 - 1.69i)T \) |
| 37 | \( 1 + (-3.63 - 4.87i)T \) |
good | 3 | \( 1 + (1.28 - 1.28i)T - 3iT^{2} \) |
| 7 | \( 1 + (-0.579 + 0.579i)T - 7iT^{2} \) |
| 11 | \( 1 - 3.64iT - 11T^{2} \) |
| 13 | \( 1 - 3.10iT - 13T^{2} \) |
| 17 | \( 1 + 8.09T + 17T^{2} \) |
| 19 | \( 1 + (-3.05 + 3.05i)T - 19iT^{2} \) |
| 23 | \( 1 + 7.79iT - 23T^{2} \) |
| 29 | \( 1 + (1.37 + 1.37i)T + 29iT^{2} \) |
| 31 | \( 1 + (-3.23 + 3.23i)T - 31iT^{2} \) |
| 41 | \( 1 - 9.31iT - 41T^{2} \) |
| 43 | \( 1 - 10.9iT - 43T^{2} \) |
| 47 | \( 1 + (4.11 - 4.11i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.446 - 0.446i)T + 53iT^{2} \) |
| 59 | \( 1 + (-6.16 + 6.16i)T - 59iT^{2} \) |
| 61 | \( 1 + (-8.71 + 8.71i)T - 61iT^{2} \) |
| 67 | \( 1 + (-2.01 - 2.01i)T + 67iT^{2} \) |
| 71 | \( 1 - 0.00151T + 71T^{2} \) |
| 73 | \( 1 + (-9.32 + 9.32i)T - 73iT^{2} \) |
| 79 | \( 1 + (0.760 - 0.760i)T - 79iT^{2} \) |
| 83 | \( 1 + (-3.16 - 3.16i)T + 83iT^{2} \) |
| 89 | \( 1 + (-5.80 - 5.80i)T + 89iT^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−11.41845066277581581445211825172, −10.99091944303629590743348984671, −9.852162492428735387638961191770, −9.408501363244654156689892485771, −7.990145191969927397518665600033, −6.70981618914162584940342364024, −6.36055283726904603025853389945, −4.78112270028893963176439011894, −4.47344278654808169290170836685, −2.36019406771387180579590747974,
0.70432887695903331489632602672, 2.02262626103259848706792541555, 3.70089143774324362716893420247, 5.36288670953957310087387848214, 5.74345618318177744399321811109, 7.05508921254849183773584715597, 8.427139723470004487249297316626, 9.074076561616839176519902406338, 10.19682599243993089709107087331, 11.20854829012298215960919993080