| L(s) = 1 | + 2i·3-s − 7-s − 9-s + 4i·13-s + 6·17-s − 2i·19-s − 2i·21-s + 5·25-s + 4i·27-s + 6i·29-s − 4·31-s + 2i·37-s − 8·39-s − 6·41-s + 8i·43-s + ⋯ |
| L(s) = 1 | + 1.15i·3-s − 0.377·7-s − 0.333·9-s + 1.10i·13-s + 1.45·17-s − 0.458i·19-s − 0.436i·21-s + 25-s + 0.769i·27-s + 1.11i·29-s − 0.718·31-s + 0.328i·37-s − 1.28·39-s − 0.937·41-s + 1.21i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.486006467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.486006467\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.534787196938082010907880759058, −9.098578028097509415707346161483, −8.147052255068964491050557395691, −7.10244484516959338926717268054, −6.42260107548125418837606558146, −5.22685477394171817252048297576, −4.73233704940919964481830423819, −3.67724563045336390228385423503, −3.05986235449912193549632571266, −1.46859891296255733224064010658,
0.58402828276682568742413669358, 1.68724307891371138195539183749, 2.88170665607781651375695803383, 3.73128267301782271183106888680, 5.13872359531302107959415524867, 5.87383896144609479138352216976, 6.66590458914084282251461327277, 7.49402247491036111358051913974, 7.969755220518055100671528487094, 8.787882362901984389768593964098
