L(s) = 1 | + (−1.18 − 1.18i)3-s + (−1.87 + 1.87i)5-s + i·7-s − 0.202i·9-s + (0.584 − 0.584i)11-s + (3.94 + 3.94i)13-s + 4.44·15-s + 1.74·17-s + (−4.19 − 4.19i)19-s + (1.18 − 1.18i)21-s − 3.04i·23-s − 2.05i·25-s + (−3.78 + 3.78i)27-s + (−4.43 − 4.43i)29-s + 7.90·31-s + ⋯ |
L(s) = 1 | + (−0.682 − 0.682i)3-s + (−0.839 + 0.839i)5-s + 0.377i·7-s − 0.0675i·9-s + (0.176 − 0.176i)11-s + (1.09 + 1.09i)13-s + 1.14·15-s + 0.424·17-s + (−0.962 − 0.962i)19-s + (0.258 − 0.258i)21-s − 0.634i·23-s − 0.411i·25-s + (−0.728 + 0.728i)27-s + (−0.823 − 0.823i)29-s + 1.42·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4213185316\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4213185316\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (1.18 + 1.18i)T + 3iT^{2} \) |
| 5 | \( 1 + (1.87 - 1.87i)T - 5iT^{2} \) |
| 11 | \( 1 + (-0.584 + 0.584i)T - 11iT^{2} \) |
| 13 | \( 1 + (-3.94 - 3.94i)T + 13iT^{2} \) |
| 17 | \( 1 - 1.74T + 17T^{2} \) |
| 19 | \( 1 + (4.19 + 4.19i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.04iT - 23T^{2} \) |
| 29 | \( 1 + (4.43 + 4.43i)T + 29iT^{2} \) |
| 31 | \( 1 - 7.90T + 31T^{2} \) |
| 37 | \( 1 + (5.87 - 5.87i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.38iT - 41T^{2} \) |
| 43 | \( 1 + (-1.73 + 1.73i)T - 43iT^{2} \) |
| 47 | \( 1 + 1.80T + 47T^{2} \) |
| 53 | \( 1 + (9.73 - 9.73i)T - 53iT^{2} \) |
| 59 | \( 1 + (4.74 - 4.74i)T - 59iT^{2} \) |
| 61 | \( 1 + (-3.10 - 3.10i)T + 61iT^{2} \) |
| 67 | \( 1 + (4.81 + 4.81i)T + 67iT^{2} \) |
| 71 | \( 1 - 1.11iT - 71T^{2} \) |
| 73 | \( 1 - 11.2iT - 73T^{2} \) |
| 79 | \( 1 + 7.61T + 79T^{2} \) |
| 83 | \( 1 + (-11.1 - 11.1i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.428iT - 89T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−9.457071942415125635140340200103, −8.677679257782821363931320248762, −7.904298963037922743794450496949, −6.89246556429860301357249988801, −6.53677473285000635078510841707, −5.86596381113386791355370777537, −4.53834157278611099801977153727, −3.71880199230646288281884967041, −2.66952031084030960944875741783, −1.30453401783437258355237629227,
0.19048193026420860520907392482, 1.53978116193755683343061644742, 3.42433808311817226462478083623, 4.04138901354047963038054524181, 4.88559832515477065136505499436, 5.59150699776799917232030460143, 6.41895248728384132016663557412, 7.73458529425065556978106969790, 8.112727630040124342936604712848, 8.943192141552287824183759100757