L(s) = 1 | + (0.904 + 0.425i)5-s + (−0.770 − 0.637i)9-s + (−0.180 − 1.91i)13-s + (−1.22 − 1.57i)17-s + (0.637 + 0.770i)25-s + (−0.340 − 0.362i)29-s + (−1.57 − 0.934i)37-s + (−0.233 − 0.0922i)41-s + (−0.425 − 0.904i)45-s + (−0.951 − 0.309i)49-s + (1.91 − 0.557i)53-s + (−1.68 + 0.666i)61-s + (0.650 − 1.80i)65-s + (0.360 + 1.61i)73-s + (0.187 + 0.982i)81-s + ⋯ |
L(s) = 1 | + (0.904 + 0.425i)5-s + (−0.770 − 0.637i)9-s + (−0.180 − 1.91i)13-s + (−1.22 − 1.57i)17-s + (0.637 + 0.770i)25-s + (−0.340 − 0.362i)29-s + (−1.57 − 0.934i)37-s + (−0.233 − 0.0922i)41-s + (−0.425 − 0.904i)45-s + (−0.951 − 0.309i)49-s + (1.91 − 0.557i)53-s + (−1.68 + 0.666i)61-s + (0.650 − 1.80i)65-s + (0.360 + 1.61i)73-s + (0.187 + 0.982i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0439 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0439 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.112608504\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.112608504\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.904 - 0.425i)T \) |
good | 3 | \( 1 + (0.770 + 0.637i)T^{2} \) |
| 7 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 11 | \( 1 + (-0.992 + 0.125i)T^{2} \) |
| 13 | \( 1 + (0.180 + 1.91i)T + (-0.982 + 0.187i)T^{2} \) |
| 17 | \( 1 + (1.22 + 1.57i)T + (-0.248 + 0.968i)T^{2} \) |
| 19 | \( 1 + (0.637 + 0.770i)T^{2} \) |
| 23 | \( 1 + (-0.684 - 0.728i)T^{2} \) |
| 29 | \( 1 + (0.340 + 0.362i)T + (-0.0627 + 0.998i)T^{2} \) |
| 31 | \( 1 + (0.968 + 0.248i)T^{2} \) |
| 37 | \( 1 + (1.57 + 0.934i)T + (0.481 + 0.876i)T^{2} \) |
| 41 | \( 1 + (0.233 + 0.0922i)T + (0.728 + 0.684i)T^{2} \) |
| 43 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 47 | \( 1 + (0.368 - 0.929i)T^{2} \) |
| 53 | \( 1 + (-1.91 + 0.557i)T + (0.844 - 0.535i)T^{2} \) |
| 59 | \( 1 + (0.425 - 0.904i)T^{2} \) |
| 61 | \( 1 + (1.68 - 0.666i)T + (0.728 - 0.684i)T^{2} \) |
| 67 | \( 1 + (-0.998 + 0.0627i)T^{2} \) |
| 71 | \( 1 + (-0.929 - 0.368i)T^{2} \) |
| 73 | \( 1 + (-0.360 - 1.61i)T + (-0.904 + 0.425i)T^{2} \) |
| 79 | \( 1 + (0.637 - 0.770i)T^{2} \) |
| 83 | \( 1 + (-0.770 + 0.637i)T^{2} \) |
| 89 | \( 1 + (-1.68 - 1.06i)T + (0.425 + 0.904i)T^{2} \) |
| 97 | \( 1 + (-1.95 - 0.0613i)T + (0.998 + 0.0627i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.670935245356631584137728087745, −7.60199337770892121189197065627, −6.96701814359599445873322334706, −6.18416975240049498803720118342, −5.46936325888064102077649216571, −4.98606595396468483822252876695, −3.58936438692619066186152882130, −2.87884832194065599491588422713, −2.19155316805167939535886148509, −0.56836580544248936518399511761,
1.81733992241110653377350178973, 2.02817979536375825319690222686, 3.38407848570269008447101557545, 4.49847630707836871438384014257, 4.93470315465420772828649275714, 6.03529669272984719113394541920, 6.39919206479778828177046974491, 7.25309960497273008311741067376, 8.337303618025485506285461060496, 8.874010955438326469863798047493