L(s) = 1 | + (1.70 − 0.292i)3-s + (2 + 2i)7-s + (2.82 − i)9-s + 5.65i·11-s + (−2.82 + 2.82i)17-s − 4i·19-s + (4 + 2.82i)21-s + (−4.24 − 4.24i)23-s + (4.53 − 2.53i)27-s + 5.65·29-s + 8·31-s + (1.65 + 9.65i)33-s + (−8 − 8i)37-s + 5.65i·41-s + (2 − 2i)43-s + ⋯ |
L(s) = 1 | + (0.985 − 0.169i)3-s + (0.755 + 0.755i)7-s + (0.942 − 0.333i)9-s + 1.70i·11-s + (−0.685 + 0.685i)17-s − 0.917i·19-s + (0.872 + 0.617i)21-s + (−0.884 − 0.884i)23-s + (0.872 − 0.487i)27-s + 1.05·29-s + 1.43·31-s + (0.288 + 1.68i)33-s + (−1.31 − 1.31i)37-s + 0.883i·41-s + (0.304 − 0.304i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.14924 + 0.437623i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.14924 + 0.437623i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.70 + 0.292i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-2 - 2i)T + 7iT^{2} \) |
| 11 | \( 1 - 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (2.82 - 2.82i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (4.24 + 4.24i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (8 + 8i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (-2 + 2i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.41 + 1.41i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.65 + 5.65i)T + 53iT^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (-6 - 6i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.3iT - 71T^{2} \) |
| 73 | \( 1 + (8 - 8i)T - 73iT^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + (9.89 + 9.89i)T + 83iT^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + (-8 - 8i)T + 97iT^{2} \) |
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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
−10.49708862711964756298045257826, −9.788281705732756957387823429899, −8.777494271323086892898375163882, −8.277128103438994246555553898084, −7.25048871863829031643520472450, −6.43561408897137392191719354693, −4.87254526941122579339200741349, −4.22177179584801935656379819514, −2.57669095875863565541452278991, −1.85214976429759715755345088730,
1.31086555829133166954700244857, 2.86690011280739985368394662212, 3.85038367217096879344964794393, 4.81949693080983703297922330155, 6.12257249549774326151727324542, 7.27331507502432617979734143967, 8.212369006360637087168124125263, 8.572840795598017921892430684499, 9.769295788967416636797011897455, 10.52245619461867597643163917101