L(s) = 1 | + (−0.386 + 2.20i)5-s + (1.51 + 1.51i)7-s + 3.92·11-s + (−3.56 + 3.56i)13-s + (1.37 − 1.37i)17-s + 4i·19-s + (5.17 − 5.17i)23-s + (−4.70 − 1.70i)25-s − 5.95·29-s + 7.12i·31-s + (−3.92 + 2.75i)35-s + (3.56 + 3.56i)37-s − 2.75·41-s + (−5.40 − 5.40i)43-s + (1.54 + 1.54i)47-s + ⋯ |
L(s) = 1 | + (−0.172 + 0.984i)5-s + (0.573 + 0.573i)7-s + 1.18·11-s + (−0.988 + 0.988i)13-s + (0.333 − 0.333i)17-s + 0.917i·19-s + (1.07 − 1.07i)23-s + (−0.940 − 0.340i)25-s − 1.10·29-s + 1.28i·31-s + (−0.663 + 0.465i)35-s + (0.585 + 0.585i)37-s − 0.430·41-s + (−0.823 − 0.823i)43-s + (0.225 + 0.225i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.579396790\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.579396790\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.386 - 2.20i)T \) |
good | 7 | \( 1 + (-1.51 - 1.51i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.92T + 11T^{2} \) |
| 13 | \( 1 + (3.56 - 3.56i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.37 + 1.37i)T - 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (-5.17 + 5.17i)T - 23iT^{2} \) |
| 29 | \( 1 + 5.95T + 29T^{2} \) |
| 31 | \( 1 - 7.12iT - 31T^{2} \) |
| 37 | \( 1 + (-3.56 - 3.56i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.75T + 41T^{2} \) |
| 43 | \( 1 + (5.40 + 5.40i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.54 - 1.54i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.81 - 1.81i)T - 53iT^{2} \) |
| 59 | \( 1 - 3.92iT - 59T^{2} \) |
| 61 | \( 1 - 13.1iT - 61T^{2} \) |
| 67 | \( 1 + (-5.40 + 5.40i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-5 - 5i)T + 73iT^{2} \) |
| 79 | \( 1 + 7.12T + 79T^{2} \) |
| 83 | \( 1 + (6.67 + 6.67i)T + 83iT^{2} \) |
| 89 | \( 1 - 18.4iT - 89T^{2} \) |
| 97 | \( 1 + (10.4 - 10.4i)T - 97iT^{2} \) |
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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
−9.731492183869682206946509888684, −8.994779691512263660297776605902, −8.191488755583294229486097892395, −7.07992882824857450195145209325, −6.77141688292286435919454343864, −5.67593864478864874922102374904, −4.68336522624002569201562832643, −3.73673708910791464966347583012, −2.66637533005350693041620762608, −1.62282356344892248075305403796,
0.65963109382441277123191666655, 1.76064895375564107206497301320, 3.31973742118200838076900491114, 4.29355357271287600228097903808, 5.03288488204353116621145814306, 5.82511874569815928388754901544, 7.09134212781612015889716296751, 7.69118838540247383971968712052, 8.454510246285937354291438484535, 9.458600721356397943432886891239