L(s) = 1 | − 3-s + i·5-s + 3i·7-s + 9-s − 3i·11-s + (−2 + 3i)13-s − i·15-s − 17-s − 8i·19-s − 3i·21-s − 3·23-s − 25-s − 27-s + 6·29-s − 10i·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447i·5-s + 1.13i·7-s + 0.333·9-s − 0.904i·11-s + (−0.554 + 0.832i)13-s − 0.258i·15-s − 0.242·17-s − 1.83i·19-s − 0.654i·21-s − 0.625·23-s − 0.200·25-s − 0.192·27-s + 1.11·29-s − 1.79i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.117680336\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.117680336\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (2 - 3i)T \) |
good | 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 + 5iT - 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 5T + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 11iT - 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 15iT - 89T^{2} \) |
| 97 | \( 1 + 5iT - 97T^{2} \) |
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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
−8.736021526128855840953301888000, −7.88185501696586054412856730126, −6.89209150808045119435277415292, −6.38261955719910837778503718568, −5.62691143084876771401305366555, −4.88017786725238940767467832699, −3.99423638422795818520451270105, −2.75366669257224742940402395993, −2.17159143454363132259971524425, −0.47152014830353985402368761462,
0.914456867938377327877454309290, 1.90228680297599406094451331378, 3.35102590633591469988416350528, 4.20114396245675254829938265592, 4.89765307384382191161466602631, 5.61602921475190339827241710227, 6.61424432096851935362557546063, 7.20634875873734986872041077410, 7.977575739374684492708958779186, 8.583205102073582068646057373769