L(s) = 1 | + (−0.707 + 0.707i)2-s + (1.32 − 1.32i)3-s − 1.00i·4-s + 1.88i·6-s + (−1.36 − 2.26i)7-s + (0.707 + 0.707i)8-s − 0.535i·9-s + 1.73·11-s + (−1.32 − 1.32i)12-s + (3.63 − 3.63i)13-s + (2.56 + 0.633i)14-s − 1.00·16-s + (−2.30 − 2.30i)17-s + (0.378 + 0.378i)18-s + 3.25·19-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.767 − 0.767i)3-s − 0.500i·4-s + 0.767i·6-s + (−0.517 − 0.855i)7-s + (0.250 + 0.250i)8-s − 0.178i·9-s + 0.522·11-s + (−0.383 − 0.383i)12-s + (1.00 − 1.00i)13-s + (0.686 + 0.169i)14-s − 0.250·16-s + (−0.558 − 0.558i)17-s + (0.0893 + 0.0893i)18-s + 0.747·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15025 - 0.552964i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15025 - 0.552964i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.36 + 2.26i)T \) |
good | 3 | \( 1 + (-1.32 + 1.32i)T - 3iT^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 + (-3.63 + 3.63i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.30 + 2.30i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.25T + 19T^{2} \) |
| 23 | \( 1 + (5.79 + 5.79i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.73iT - 29T^{2} \) |
| 31 | \( 1 + 8.89iT - 31T^{2} \) |
| 37 | \( 1 + (1.55 - 1.55i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.64iT - 41T^{2} \) |
| 43 | \( 1 + (-2.44 - 2.44i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.29 - 6.29i)T + 47iT^{2} \) |
| 53 | \( 1 + (-10.0 - 10.0i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6.51iT - 61T^{2} \) |
| 67 | \( 1 + (7.02 - 7.02i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.19T + 71T^{2} \) |
| 73 | \( 1 + (7.62 - 7.62i)T - 73iT^{2} \) |
| 79 | \( 1 - 2iT - 79T^{2} \) |
| 83 | \( 1 + (3.98 - 3.98i)T - 83iT^{2} \) |
| 89 | \( 1 + 5.64T + 89T^{2} \) |
| 97 | \( 1 + (7.26 + 7.26i)T + 97iT^{2} \) |
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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
−11.14862133642250396477372699231, −10.30215154731658239230808228763, −9.306826636727519012880897975256, −8.354408276441873511426561932617, −7.64685725726034740261558646651, −6.81765254961178161230678220727, −5.86402433978030179817746104898, −4.18310520795319984365747778476, −2.76622324248721604029041624194, −1.05251672814230000090785040954,
1.95938144155557047547815154130, 3.40844448384558025957640123571, 4.07108015676076046146059840254, 5.79989599612519449478263935890, 6.93407348358533023239615768762, 8.450294955287777831925240143362, 8.951133843997797781322380715470, 9.602061356163818543197417696628, 10.46567160564740665168434907427, 11.68454587763195825872525749962