L(s) = 1 | + 3-s − 5-s − 3.73·7-s + 9-s + 4.84·11-s − 2.84·13-s − 15-s + 0.890·17-s + 6.84·19-s − 3.73·21-s + 23-s + 25-s + 27-s − 0.890·29-s + 7.73·31-s + 4.84·33-s + 3.73·35-s − 1.95·37-s − 2.84·39-s + 12.3·41-s − 3.47·43-s − 45-s − 6.62·47-s + 6.95·49-s + 0.890·51-s + 12.3·53-s − 4.84·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.41·7-s + 0.333·9-s + 1.46·11-s − 0.788·13-s − 0.258·15-s + 0.216·17-s + 1.57·19-s − 0.815·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s − 0.165·29-s + 1.38·31-s + 0.843·33-s + 0.631·35-s − 0.321·37-s − 0.455·39-s + 1.93·41-s − 0.529·43-s − 0.149·45-s − 0.966·47-s + 0.993·49-s + 0.124·51-s + 1.69·53-s − 0.653·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.750599456\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.750599456\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 3.73T + 7T^{2} \) |
| 11 | \( 1 - 4.84T + 11T^{2} \) |
| 13 | \( 1 + 2.84T + 13T^{2} \) |
| 17 | \( 1 - 0.890T + 17T^{2} \) |
| 19 | \( 1 - 6.84T + 19T^{2} \) |
| 29 | \( 1 + 0.890T + 29T^{2} \) |
| 31 | \( 1 - 7.73T + 31T^{2} \) |
| 37 | \( 1 + 1.95T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 + 3.47T + 43T^{2} \) |
| 47 | \( 1 + 6.62T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 0.890T + 59T^{2} \) |
| 61 | \( 1 - 8.62T + 61T^{2} \) |
| 67 | \( 1 - 7.73T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 + 17.2T + 97T^{2} \) |
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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
−9.689468344054512375997941584647, −8.932435827041407141097634525861, −7.953559415058402442644368898593, −7.04304329945705091762746347712, −6.57819038218311113684043615644, −5.43287411975762953409106091050, −4.19296708160366480358094398543, −3.45889390362862916763654605637, −2.65190881754065178858108018149, −0.960568129185171567134975236221,
0.960568129185171567134975236221, 2.65190881754065178858108018149, 3.45889390362862916763654605637, 4.19296708160366480358094398543, 5.43287411975762953409106091050, 6.57819038218311113684043615644, 7.04304329945705091762746347712, 7.953559415058402442644368898593, 8.932435827041407141097634525861, 9.689468344054512375997941584647