L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + (−0.597 − 2.15i)5-s + 1.00·6-s + (1.47 − 1.47i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (−1.10 + 1.94i)10-s − 0.912i·11-s + (−0.707 − 0.707i)12-s + (−1.90 + 1.90i)13-s − 2.09·14-s + (1.94 + 1.10i)15-s − 1.00·16-s + (2.19 − 2.19i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (−0.267 − 0.963i)5-s + 0.408·6-s + (0.559 − 0.559i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.348 + 0.615i)10-s − 0.275i·11-s + (−0.204 − 0.204i)12-s + (−0.528 + 0.528i)13-s − 0.559·14-s + (0.502 + 0.284i)15-s − 0.250·16-s + (0.531 − 0.531i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 + 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.151402 - 0.587073i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.151402 - 0.587073i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.597 + 2.15i)T \) |
| 23 | \( 1 + (-4.55 - 1.50i)T \) |
good | 7 | \( 1 + (-1.47 + 1.47i)T - 7iT^{2} \) |
| 11 | \( 1 + 0.912iT - 11T^{2} \) |
| 13 | \( 1 + (1.90 - 1.90i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.19 + 2.19i)T - 17iT^{2} \) |
| 19 | \( 1 + 5.16T + 19T^{2} \) |
| 29 | \( 1 + 8.27iT - 29T^{2} \) |
| 31 | \( 1 + 5.45T + 31T^{2} \) |
| 37 | \( 1 + (-3.73 + 3.73i)T - 37iT^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + (0.0783 + 0.0783i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.92 + 4.92i)T + 47iT^{2} \) |
| 53 | \( 1 + (8.84 + 8.84i)T + 53iT^{2} \) |
| 59 | \( 1 + 5.65iT - 59T^{2} \) |
| 61 | \( 1 + 9.70iT - 61T^{2} \) |
| 67 | \( 1 + (-5.22 + 5.22i)T - 67iT^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + (8.97 - 8.97i)T - 73iT^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 + (-8.86 - 8.86i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.837T + 89T^{2} \) |
| 97 | \( 1 + (-0.302 + 0.302i)T - 97iT^{2} \) |
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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
−9.998200773200970595199542962899, −9.411289532994847953138326758349, −8.471430284960935776496283186702, −7.75192476412385808786163550428, −6.67816954162642294917110707639, −5.26524801115278079889155670679, −4.53536939661288217402618353015, −3.59538614195039222646625447968, −1.84790052354229202522120862325, −0.39539171548103665402282213837,
1.73700026320025361941504311823, 3.05166588061420880787533583233, 4.69543426447121286949761002826, 5.65209484091128846802706826554, 6.58874825710855076272011164552, 7.29143127735908226660528540011, 8.107885663632103863991693980057, 8.903897194513449565434067803463, 10.17186408098877860110730647994, 10.70474134722440224754987845612