L(s) = 1 | + (1.24 − 1.24i)2-s + (0.474 − 1.66i)3-s − 1.09i·4-s + (−1.67 + 1.48i)5-s + (−1.48 − 2.66i)6-s + (0.707 + 0.707i)7-s + (1.12 + 1.12i)8-s + (−2.54 − 1.58i)9-s + (−0.241 + 3.92i)10-s − 1.55i·11-s + (−1.82 − 0.520i)12-s + (−4.50 + 4.50i)13-s + 1.75·14-s + (1.67 + 3.49i)15-s + 4.99·16-s + (2.13 − 2.13i)17-s + ⋯ |
L(s) = 1 | + (0.879 − 0.879i)2-s + (0.274 − 0.961i)3-s − 0.547i·4-s + (−0.749 + 0.662i)5-s + (−0.604 − 1.08i)6-s + (0.267 + 0.267i)7-s + (0.397 + 0.397i)8-s + (−0.849 − 0.527i)9-s + (−0.0763 + 1.24i)10-s − 0.468i·11-s + (−0.526 − 0.150i)12-s + (−1.25 + 1.25i)13-s + 0.470·14-s + (0.431 + 0.902i)15-s + 1.24·16-s + (0.517 − 0.517i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16612 - 0.939365i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16612 - 0.939365i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.474 + 1.66i)T \) |
| 5 | \( 1 + (1.67 - 1.48i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 2 | \( 1 + (-1.24 + 1.24i)T - 2iT^{2} \) |
| 11 | \( 1 + 1.55iT - 11T^{2} \) |
| 13 | \( 1 + (4.50 - 4.50i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.13 + 2.13i)T - 17iT^{2} \) |
| 19 | \( 1 + 4.20iT - 19T^{2} \) |
| 23 | \( 1 + (-3.76 - 3.76i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.97T + 29T^{2} \) |
| 31 | \( 1 + 5.79T + 31T^{2} \) |
| 37 | \( 1 + (1.23 + 1.23i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.68iT - 41T^{2} \) |
| 43 | \( 1 + (2.09 - 2.09i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.0358 + 0.0358i)T - 47iT^{2} \) |
| 53 | \( 1 + (4.30 + 4.30i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.93T + 59T^{2} \) |
| 61 | \( 1 - 3.31T + 61T^{2} \) |
| 67 | \( 1 + (-1.71 - 1.71i)T + 67iT^{2} \) |
| 71 | \( 1 + 5.73iT - 71T^{2} \) |
| 73 | \( 1 + (-7.26 + 7.26i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.59iT - 79T^{2} \) |
| 83 | \( 1 + (-12.2 - 12.2i)T + 83iT^{2} \) |
| 89 | \( 1 - 1.35T + 89T^{2} \) |
| 97 | \( 1 + (-10.9 - 10.9i)T + 97iT^{2} \) |
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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
−13.46100140611411562445280710230, −12.36650912019026374253030226138, −11.64330368309536334413975683115, −11.08433615196800983346104918824, −9.230618809944314414276477358183, −7.76883423147478770854487089807, −6.93442536143450288559938280710, −5.10320841135499756222383655036, −3.48264625043685039198652010157, −2.29182093906004056159753011241,
3.62426219879888995173833961014, 4.75459625624418050483002794345, 5.50311925122085767382255557098, 7.40790611156993511909147527596, 8.224024431083723839592141786618, 9.768175695324107552670170630021, 10.70411479180597834326504765982, 12.27819055868597843146787171820, 13.05474687811661845852867627955, 14.60336239272296355329143865815