L(s) = 1 | + i·2-s − 0.302i·3-s − 4-s + 0.302·6-s − 3.30i·7-s − i·8-s + 2.90·9-s − 1.69·11-s + 0.302i·12-s + 3.30i·13-s + 3.30·14-s + 16-s − 6.90i·17-s + 2.90i·18-s − 5.90·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.174i·3-s − 0.5·4-s + 0.123·6-s − 1.24i·7-s − 0.353i·8-s + 0.969·9-s − 0.511·11-s + 0.0874i·12-s + 0.916i·13-s + 0.882·14-s + 0.250·16-s − 1.67i·17-s + 0.685i·18-s − 1.35·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.175586192\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.175586192\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 + 0.302iT - 3T^{2} \) |
| 7 | \( 1 + 3.30iT - 7T^{2} \) |
| 11 | \( 1 + 1.69T + 11T^{2} \) |
| 13 | \( 1 - 3.30iT - 13T^{2} \) |
| 17 | \( 1 + 6.90iT - 17T^{2} \) |
| 19 | \( 1 + 5.90T + 19T^{2} \) |
| 29 | \( 1 - 2.60T + 29T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 0.908T + 41T^{2} \) |
| 43 | \( 1 + 9.21iT - 43T^{2} \) |
| 47 | \( 1 - 2.60iT - 47T^{2} \) |
| 53 | \( 1 + 11.2iT - 53T^{2} \) |
| 59 | \( 1 - 3.39T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 + 5.81iT - 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 11.2iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 6.30iT - 97T^{2} \) |
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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
−9.589569004125278615822060131527, −8.798617459542333043043557003225, −7.72114658416664675656374394818, −7.05069534485786093392430640588, −6.73309595043211026452635701941, −5.35727916230254136361379841954, −4.43874567944835561252862202381, −3.81331076608123693239281415861, −2.10826978460110325798968108821, −0.51602776230605901476758683100,
1.59339858275510356301425609484, 2.58828813060652595344920831582, 3.69966725540277802712283340580, 4.66665806121396910012215287151, 5.62232521436253832620673770507, 6.39312512637828103682891808541, 7.76535527669204986831522020539, 8.462016886086835166560370080367, 9.157517242754610773777171359945, 10.23671620465695327450840818222