| L(s) = 1 | − 2.66i·2-s − i·3-s − 5.12·4-s − 2.66·6-s + 0.454i·7-s + 8.33i·8-s − 9-s − 1.45·11-s + 5.12i·12-s + 3.33i·13-s + 1.21·14-s + 12.0·16-s + 2.66i·18-s − 19-s + 0.454·21-s + 3.88i·22-s + ⋯ |
| L(s) = 1 | − 1.88i·2-s − 0.577i·3-s − 2.56·4-s − 1.08·6-s + 0.171i·7-s + 2.94i·8-s − 0.333·9-s − 0.438·11-s + 1.47i·12-s + 0.925i·13-s + 0.324·14-s + 3.00·16-s + 0.629i·18-s − 0.229·19-s + 0.0992·21-s + 0.827i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8845368310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8845368310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.66iT - 2T^{2} \) |
| 7 | \( 1 - 0.454iT - 7T^{2} \) |
| 11 | \( 1 + 1.45T + 11T^{2} \) |
| 13 | \( 1 - 3.33iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 23 | \( 1 - 6.79iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 8.79T + 31T^{2} \) |
| 37 | \( 1 - 10.2iT - 37T^{2} \) |
| 41 | \( 1 - 3.97T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 2.42iT - 47T^{2} \) |
| 53 | \( 1 + 13.6iT - 53T^{2} \) |
| 59 | \( 1 + 7.54T + 59T^{2} \) |
| 61 | \( 1 + 3.90T + 61T^{2} \) |
| 67 | \( 1 - 14.1iT - 67T^{2} \) |
| 71 | \( 1 + 9.13T + 71T^{2} \) |
| 73 | \( 1 - 11.6iT - 73T^{2} \) |
| 79 | \( 1 + 8.97T + 79T^{2} \) |
| 83 | \( 1 - 4.54iT - 83T^{2} \) |
| 89 | \( 1 - 0.0901T + 89T^{2} \) |
| 97 | \( 1 - 12.0iT - 97T^{2} \) |
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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
−9.655774119351435592507217709654, −8.811904994438579135230287028386, −8.174210079608733167632962910649, −7.09266265750294915523519082760, −5.84892812449078544469849904446, −4.89973613435867959850572977958, −3.99270292245372902438077201640, −2.99430054008238813575431285305, −2.12650462002888581291474442195, −1.17900201547043981912504346540,
0.41212784880856727998113936686, 2.95244250507849088387809062583, 4.25727924540442761406113375008, 4.75663136546294923284337434527, 5.80297176196154308480478381436, 6.22929945422030144585613772295, 7.38877108903161828876367716181, 7.898052360212946602181354039720, 8.705499582381530869938391626219, 9.329897160379851314215730318620
