| L(s) = 1 | + 2i·5-s − 7-s + 3·9-s + 4i·11-s − 2i·13-s − 6·17-s + 8i·19-s + 25-s − 6i·29-s − 8·31-s − 2i·35-s − 2i·37-s − 2·41-s + 4i·43-s + 6i·45-s + ⋯ |
| L(s) = 1 | + 0.894i·5-s − 0.377·7-s + 9-s + 1.20i·11-s − 0.554i·13-s − 1.45·17-s + 1.83i·19-s + 0.200·25-s − 1.11i·29-s − 1.43·31-s − 0.338i·35-s − 0.328i·37-s − 0.312·41-s + 0.609i·43-s + 0.894i·45-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
| \(L(1)\) |
\(\approx\) |
\(1.148105728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.148105728\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3T^{2} \) |
| 5 | \( 1 - 2iT - 5T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.745836656439833316282704051939, −8.903712408776508411217315203105, −7.70571146092749935795184798176, −7.25214299997128106071549019140, −6.50588879532471922078838354857, −5.68784206126345907746742174977, −4.42345508652110506433275110732, −3.83905909196373736255552550968, −2.61355728361102460213543875210, −1.67703211091000580964232125707,
0.41698006543836759305244622919, 1.69115921376259029597674795708, 2.98388876822502447763400489625, 4.13262069038847401481089269234, 4.78320428605617232983467798508, 5.67284219814268850127895572559, 6.84466491763944435938810317878, 7.11196304633871858201230720088, 8.582266227698537095806042349993, 8.890938324018158262513287189505
