L(s) = 1 | − 2i·3-s + 1.73i·5-s + i·7-s − 9-s − 5.19·11-s + 4·13-s + 3.46·15-s + 3·17-s + (−1.73 + 4i)19-s + 2·21-s + 2.00·25-s − 4i·27-s + 6·29-s + 10.3·31-s + 10.3i·33-s + ⋯ |
L(s) = 1 | − 1.15i·3-s + 0.774i·5-s + 0.377i·7-s − 0.333·9-s − 1.56·11-s + 1.10·13-s + 0.894·15-s + 0.727·17-s + (−0.397 + 0.917i)19-s + 0.436·21-s + 0.400·25-s − 0.769i·27-s + 1.11·29-s + 1.86·31-s + 1.80i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.632564572\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.632564572\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.73 - 4i)T \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 5.19T + 43T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 1.73iT - 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 + 3.46iT - 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−9.949770029042992085401236253074, −8.497520786281155899362514599706, −8.029304858588916860570874271097, −7.30979380752012034527707752053, −6.32442775985774238655683943201, −5.89725232891791439454123346175, −4.61606933427779664118107664219, −3.15570216796774307805720613851, −2.43832786147612692722391104901, −1.12551900768180990217582159057,
0.878931248515100674979406216293, 2.70329741715616803652970730915, 3.75931286424274109137801982329, 4.73617757741179802522364513615, 5.14034891632527265725200915838, 6.25373749940938337028082815544, 7.43107567088914801078088881719, 8.415512130000592698239238352113, 8.851616233902046737864257077515, 9.955897386244698434356471952223