| L(s) = 1 | − i·2-s + 4-s − 3i·7-s − 3i·8-s + 11-s − i·13-s − 3·14-s − 16-s + 5i·17-s + 8·19-s − i·22-s − 26-s − 3i·28-s + 29-s + 3·31-s − 5i·32-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.5·4-s − 1.13i·7-s − 1.06i·8-s + 0.301·11-s − 0.277i·13-s − 0.801·14-s − 0.250·16-s + 1.21i·17-s + 1.83·19-s − 0.213i·22-s − 0.196·26-s − 0.566i·28-s + 0.185·29-s + 0.538·31-s − 0.883i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.366312181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.366312181\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 17 | \( 1 - 5iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 11iT - 47T^{2} \) |
| 53 | \( 1 + 11iT - 53T^{2} \) |
| 59 | \( 1 - 5T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 3iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.536075188297016073188684519429, −7.52388132689430405389929909535, −7.20639364535190907230784114761, −6.27608067808681876030254641108, −5.47958233966106311363163820785, −4.22787468512535553114394299575, −3.66455015448418708207104958393, −2.83034311294789895943277843506, −1.61841926705566144331807660126, −0.802292087908999708843012353610,
1.34254585714859250928241258766, 2.59271083236778454334353384430, 3.11525308982917661605294951285, 4.63755240687546423405564849878, 5.36364189423343105860045754700, 5.94751400576268089224502979778, 6.78764447935402384712102073594, 7.39234293722107268220631783082, 8.139830759317546178029753897431, 8.923448782788141767812499077872
