| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − 4i·7-s − i·8-s − 9-s + 4·11-s + i·12-s + 4·14-s + 16-s + 2i·17-s − i·18-s − 19-s − 4·21-s + 4i·22-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.51i·7-s − 0.353i·8-s − 0.333·9-s + 1.20·11-s + 0.288i·12-s + 1.06·14-s + 0.250·16-s + 0.485i·17-s − 0.235i·18-s − 0.229·19-s − 0.872·21-s + 0.852i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.681283141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.681283141\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 10iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 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
−8.499159188035330344554895026198, −7.76252151051415262120386576000, −7.02454506213722211429463104947, −6.57048264420955247487545925404, −5.89364909250228128511606023411, −4.57608245003361116656499426239, −4.14161021826269028708360943105, −3.10796775056040567510113070050, −1.54305139657801473633269937797, −0.61976720010593364723207997820,
1.23801903575987554825865432874, 2.49789346200610138262890858836, 3.07568648604847498666448083071, 4.23663863485613693660125827639, 4.79139275687943335467170495903, 5.88314331476774635472977651492, 6.28061167244187554116487598269, 7.60799564017764026109824089240, 8.513403679622155291112088585860, 9.172678692651458236724416013991
