L(s) = 1 | + (−0.951 + 0.309i)3-s + (0.809 + 0.587i)4-s + (−0.363 + 0.5i)7-s + (0.809 − 0.587i)9-s + (−0.951 − 0.309i)12-s + (1.53 − 0.5i)13-s + (0.309 + 0.951i)16-s + (0.5 − 1.53i)19-s + (0.190 − 0.587i)21-s + (−0.587 + 0.809i)27-s + (−0.587 + 0.190i)28-s + (0.309 + 0.951i)31-s + 0.999·36-s − 0.618i·37-s + (−1.30 + 0.951i)39-s + ⋯ |
L(s) = 1 | + (−0.951 + 0.309i)3-s + (0.809 + 0.587i)4-s + (−0.363 + 0.5i)7-s + (0.809 − 0.587i)9-s + (−0.951 − 0.309i)12-s + (1.53 − 0.5i)13-s + (0.309 + 0.951i)16-s + (0.5 − 1.53i)19-s + (0.190 − 0.587i)21-s + (−0.587 + 0.809i)27-s + (−0.587 + 0.190i)28-s + (0.309 + 0.951i)31-s + 0.999·36-s − 0.618i·37-s + (−1.30 + 0.951i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.128285411\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.128285411\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
good | 2 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 11 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + 0.618iT - T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 47 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + 1.61T + T^{2} \) |
| 67 | \( 1 - 1.61iT - T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
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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
−9.179796455557098373731146323925, −8.632784803956662265705333615563, −7.55790280386735152622294973227, −6.84373304754804633914751420547, −6.14759413117348462186657068653, −5.57408654772839290300781711997, −4.49027853399581128787696237266, −3.50484983100679206506009341463, −2.73209093115400169973549414127, −1.24186205097751753041479317853,
1.08083021053915327017412731680, 1.89489870167157514192949109507, 3.38532630698478565836255968000, 4.29705687549486583249880787815, 5.44034998160320691537535399385, 6.14707861472783468819233907659, 6.47394401467577517365928186191, 7.41524985481172616865764171258, 8.042172783254343954914870457032, 9.269120465279385617354408022752