L(s) = 1 | − i·2-s − 4-s + i·8-s + 3·9-s − 4·11-s − 2i·13-s + 16-s − 2i·17-s − 3i·18-s + 4·19-s + 4i·22-s − 2·26-s + 6·29-s − 4·31-s − i·32-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.353i·8-s + 9-s − 1.20·11-s − 0.554i·13-s + 0.250·16-s − 0.485i·17-s − 0.707i·18-s + 0.917·19-s + 0.852i·22-s − 0.392·26-s + 1.11·29-s − 0.718·31-s − 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.433572548\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.433572548\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 37 | \( 1 + iT \) |
good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−9.145902878468978505721214128673, −8.217564159640307019973250620605, −7.53063721972286753901596431189, −6.74158224590255811125427061575, −5.38479070187470452740184433098, −4.98216256622746945432233063758, −3.82612923650194385924698859474, −2.95897255745322882044362635179, −1.93306319380909118376459116832, −0.58709966963459215480189321494,
1.25159062099686705601434889498, 2.66604041419969725124253779400, 3.87851995725608663304788329540, 4.72757264560054026652267358390, 5.46480065850068405645596287034, 6.40953919155191372822760475135, 7.21849460602420205191808894740, 7.77494176285693375285206165599, 8.583453911494714143430840010870, 9.468383275079430406733088322109