L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + i·7-s + i·8-s − i·12-s + i·13-s + 14-s + 16-s − i·17-s − 21-s − 24-s + 26-s + i·27-s − i·28-s + ⋯ |
L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + i·7-s + i·8-s − i·12-s + i·13-s + 14-s + 16-s − i·17-s − 21-s − 24-s + 26-s + i·27-s − i·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9686447704\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9686447704\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 3 | \( 1 - iT - T^{2} \) |
| 7 | \( 1 - iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 17 | \( 1 + iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 2T + T^{2} \) |
| 37 | \( 1 - iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT - T^{2} \) |
| 47 | \( 1 - iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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
−9.309858786027913465307337564615, −8.965143348731152654901595631742, −7.962780757441523473655793668422, −6.90802042933717720255091797914, −5.73485165838452192604891428599, −5.01437435126731726376976657464, −4.38710127369997047439325810473, −3.52232375631695779717175110278, −2.64874601056027250164842693353, −1.62874756083702419636601617607,
0.66400420368426980752917054715, 1.87403535638343678969713244945, 3.53339647873019745774335370303, 4.14127237860610942799375375590, 5.32910483653826253440168043965, 5.95533086980217124120116354545, 6.91302117050544894852730875095, 7.28887371156706214148821550113, 7.929221259602073935583366387689, 8.575401925430477705499808454035