L(s) = 1 | + (0.987 − 0.156i)2-s + (0.951 − 0.309i)4-s + (−0.453 − 0.891i)5-s + (0.891 − 0.453i)8-s + (−0.587 − 0.809i)10-s + (−0.142 + 0.896i)13-s + (0.809 − 0.587i)16-s + (−0.533 − 1.04i)17-s + (−0.707 − 0.707i)20-s + (−0.587 + 0.809i)25-s + 0.907i·26-s + (0.610 + 1.87i)29-s + (0.707 − 0.707i)32-s + (−0.690 − 0.951i)34-s + (−0.309 − 0.0489i)37-s + ⋯ |
L(s) = 1 | + (0.987 − 0.156i)2-s + (0.951 − 0.309i)4-s + (−0.453 − 0.891i)5-s + (0.891 − 0.453i)8-s + (−0.587 − 0.809i)10-s + (−0.142 + 0.896i)13-s + (0.809 − 0.587i)16-s + (−0.533 − 1.04i)17-s + (−0.707 − 0.707i)20-s + (−0.587 + 0.809i)25-s + 0.907i·26-s + (0.610 + 1.87i)29-s + (0.707 − 0.707i)32-s + (−0.690 − 0.951i)34-s + (−0.309 − 0.0489i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.678575946\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.678575946\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.987 + 0.156i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.453 + 0.891i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.142 - 0.896i)T + (-0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (0.533 + 1.04i)T + (-0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 29 | \( 1 + (-0.610 - 1.87i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.0489i)T + (0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (1.04 + 1.44i)T + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (0.863 - 1.69i)T + (-0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 71 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (1.76 - 0.278i)T + (0.951 - 0.309i)T^{2} \) |
| 79 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + (-0.253 - 0.183i)T + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.896 + 1.76i)T + (-0.587 - 0.809i)T^{2} \) |
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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
−10.43217491404989629776185589054, −9.305020314635783967264084711832, −8.629202309632283597076948980986, −7.37004033163099433186580630751, −6.82342914285369257813225379827, −5.57078962052024463958003650785, −4.80071572186294648993526789698, −4.09832610009877531153763933818, −2.91594584388946809654208969471, −1.52614360141319530145078660978,
2.17929036591852459450067450580, 3.22932875204371312786881908952, 4.04742927709698840884975793573, 5.10635235600885901129405361651, 6.23184802219006896851038236338, 6.71119540166231237884624369981, 7.87164508951141728687826455595, 8.268584491802090880412261008462, 9.977947496226990216364125933939, 10.50881760726300515051965708608