| L(s) = 1 | − i·2-s − 4-s + (−2 − i)5-s + i·8-s + (−1 + 2i)10-s + 4·11-s + 4i·13-s + 16-s − 6i·17-s + 19-s + (2 + i)20-s − 4i·22-s + (3 + 4i)25-s + 4·26-s − 2·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.894 − 0.447i)5-s + 0.353i·8-s + (−0.316 + 0.632i)10-s + 1.20·11-s + 1.10i·13-s + 0.250·16-s − 1.45i·17-s + 0.229·19-s + (0.447 + 0.223i)20-s − 0.852i·22-s + (0.600 + 0.800i)25-s + 0.784·26-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.255632201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.255632201\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 14iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 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
−9.292024342077516206329770052315, −8.535367432913759626836397428952, −7.46071392701930798830936154448, −6.88956734081434786106101238285, −5.66712805410295926267963157100, −4.58029228141831859332207397481, −4.08181108724219159152351296004, −3.13051187884498796533339256776, −1.81626712317735118204374675795, −0.58515639726989476720516646524,
1.13232696657335121694437892039, 2.94209847647220260044219177673, 3.87982933375635799366194039024, 4.50172493008255613495263289013, 5.85263583998635103419103169491, 6.30341858174313776356701208912, 7.36732610777691659276556120731, 7.83842740388649712558927411850, 8.647348584629239805046654344710, 9.374370911423795205274291342450
