| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + i·7-s + i·8-s − 9-s − 3·11-s + i·12-s + 4i·13-s + 14-s + 16-s + i·17-s + i·18-s + 5·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.377i·7-s + 0.353i·8-s − 0.333·9-s − 0.904·11-s + 0.288i·12-s + 1.10i·13-s + 0.267·14-s + 0.250·16-s + 0.242i·17-s + 0.235i·18-s + 1.14·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.377973371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.377973371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 - iT \) |
| good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 + 7iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 11iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 15T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.750665164041458623983831845462, −7.990158316841204488947389683822, −7.17862736801945245626673622924, −6.34975667134583752122398761345, −5.43143616327414868446114459656, −4.68388897049237560413442757728, −3.63809958627429807223981996121, −2.58568942522214367678035731820, −1.95033174118508858370398304990, −0.56614619185317674264079813213,
0.988243288867752142068473903686, 2.81238748262843188628141558196, 3.52363024777818826798405531636, 4.58256089390680482103131667670, 5.40710695070715015431231173032, 5.76003013289323366722328892513, 7.02104696732293284844404007740, 7.66821475429571238332984014363, 8.183956694854301605070786512292, 9.167407619902233046494222138902
