L(s) = 1 | + (−1.34 + 1.78i)5-s + (−2.90 + 2.90i)7-s − 2.15i·11-s + (−1.90 − 1.90i)13-s + (−0.974 − 0.974i)17-s − 1.80i·19-s + (−0.879 + 0.879i)23-s + (−1.37 − 4.80i)25-s + 0.743·29-s − 7.05·31-s + (−1.27 − 9.09i)35-s + (8.33 − 8.33i)37-s + 7.67i·41-s + (3.05 + 3.05i)43-s + (7.54 + 7.54i)47-s + ⋯ |
L(s) = 1 | + (−0.601 + 0.798i)5-s + (−1.09 + 1.09i)7-s − 0.650i·11-s + (−0.527 − 0.527i)13-s + (−0.236 − 0.236i)17-s − 0.414i·19-s + (−0.183 + 0.183i)23-s + (−0.275 − 0.961i)25-s + 0.137·29-s − 1.26·31-s + (−0.215 − 1.53i)35-s + (1.36 − 1.36i)37-s + 1.19i·41-s + (0.465 + 0.465i)43-s + (1.09 + 1.09i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8245932726\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8245932726\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.34 - 1.78i)T \) |
good | 7 | \( 1 + (2.90 - 2.90i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.15iT - 11T^{2} \) |
| 13 | \( 1 + (1.90 + 1.90i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.974 + 0.974i)T + 17iT^{2} \) |
| 19 | \( 1 + 1.80iT - 19T^{2} \) |
| 23 | \( 1 + (0.879 - 0.879i)T - 23iT^{2} \) |
| 29 | \( 1 - 0.743T + 29T^{2} \) |
| 31 | \( 1 + 7.05T + 31T^{2} \) |
| 37 | \( 1 + (-8.33 + 8.33i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.67iT - 41T^{2} \) |
| 43 | \( 1 + (-3.05 - 3.05i)T + 43iT^{2} \) |
| 47 | \( 1 + (-7.54 - 7.54i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.08 + 5.08i)T - 53iT^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 4.56T + 61T^{2} \) |
| 67 | \( 1 + (-7.05 + 7.05i)T - 67iT^{2} \) |
| 71 | \( 1 - 12.5iT - 71T^{2} \) |
| 73 | \( 1 + (-3.62 - 3.62i)T + 73iT^{2} \) |
| 79 | \( 1 + 15.0iT - 79T^{2} \) |
| 83 | \( 1 + (-7.81 + 7.81i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.40T + 89T^{2} \) |
| 97 | \( 1 + (-6.67 + 6.67i)T - 97iT^{2} \) |
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
−8.850132021467758631798920689263, −7.78420656567533981900974631758, −7.31117156099672104004623907139, −6.18171325419171112504874946728, −5.98113064897223576823733353256, −4.82442729503513348606354502379, −3.70530474191502799967371370688, −2.95927200409838408272207885990, −2.37911175976318953348095351973, −0.36393880354218971288828420471,
0.796451132832484406520535147245, 2.11418098606563546942470525430, 3.47077425705783387002874956852, 4.10074224516692982877382098497, 4.74132378892878990762227304381, 5.80367335356313611501051195445, 6.75349990737428814440735651438, 7.33655357048798496489102744678, 7.944097474091828461685749968890, 9.016672969008596333920463953694