L(s) = 1 | + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (0.587 − 0.809i)9-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s − 18-s + (0.951 + 0.309i)22-s + 0.618·23-s + (0.951 + 0.309i)25-s + i·28-s + (−0.142 + 0.278i)29-s + i·32-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (0.587 − 0.809i)9-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s − 18-s + (0.951 + 0.309i)22-s + 0.618·23-s + (0.951 + 0.309i)25-s + i·28-s + (−0.142 + 0.278i)29-s + i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8696191048\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8696191048\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.587 + 0.809i)T \) |
| 7 | \( 1 + (-0.951 + 0.309i)T \) |
| 11 | \( 1 + (0.809 - 0.587i)T \) |
good | 3 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 5 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 13 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 23 | \( 1 - 0.618T + T^{2} \) |
| 29 | \( 1 + (0.142 - 0.278i)T + (-0.587 - 0.809i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.412 + 0.809i)T + (-0.587 - 0.809i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + (0.221 + 0.221i)T + iT^{2} \) |
| 47 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 + 1.95i)T + (-0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 61 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 67 | \( 1 + (-1.39 - 1.39i)T + iT^{2} \) |
| 71 | \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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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
−9.892133781887678324406652424948, −9.039576099601195083197613489448, −8.298486901505559175747104190833, −7.41014669612434352237077394414, −6.87280662863395828967984567957, −5.25391967191222915493546155765, −4.46717402257202262098908928124, −3.51814801174464559447257241150, −2.30873762560547988739480324761, −1.15483492307582695350088833166,
1.38508010196545064757140024048, 2.64905461904732717544060496647, 4.47553083521713394457158767204, 5.06141650716783122140649810129, 5.87203676626175057478610839502, 6.94137601844431376222509016605, 7.78482059590272674787691229809, 8.243804352976102730709522775275, 9.035954489643240916315619090903, 10.00012532719180120383757391770