| L(s) = 1 | + 3-s + 3.52i·5-s − i·7-s + 9-s + 0.903i·11-s + (0.311 − 3.59i)13-s + 3.52i·15-s − 5.80·17-s + 3.05i·19-s − i·21-s + 2.42·23-s − 7.42·25-s + 27-s − 6.85·29-s + 2.75i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.57i·5-s − 0.377i·7-s + 0.333·9-s + 0.272i·11-s + (0.0862 − 0.996i)13-s + 0.910i·15-s − 1.40·17-s + 0.699i·19-s − 0.218i·21-s + 0.506·23-s − 1.48·25-s + 0.192·27-s − 1.27·29-s + 0.494i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8919257278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8919257278\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (-0.311 + 3.59i)T \) |
| good | 5 | \( 1 - 3.52iT - 5T^{2} \) |
| 11 | \( 1 - 0.903iT - 11T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 19 | \( 1 - 3.05iT - 19T^{2} \) |
| 23 | \( 1 - 2.42T + 23T^{2} \) |
| 29 | \( 1 + 6.85T + 29T^{2} \) |
| 31 | \( 1 - 2.75iT - 31T^{2} \) |
| 37 | \( 1 - 3.05iT - 37T^{2} \) |
| 41 | \( 1 - 0.474iT - 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 1.09iT - 47T^{2} \) |
| 53 | \( 1 + 0.755T + 53T^{2} \) |
| 59 | \( 1 - 7.76iT - 59T^{2} \) |
| 61 | \( 1 + 7.37T + 61T^{2} \) |
| 67 | \( 1 - 0.949iT - 67T^{2} \) |
| 71 | \( 1 - 7.09iT - 71T^{2} \) |
| 73 | \( 1 + 14.2iT - 73T^{2} \) |
| 79 | \( 1 + 4.13T + 79T^{2} \) |
| 83 | \( 1 - 10.5iT - 83T^{2} \) |
| 89 | \( 1 - 14.1iT - 89T^{2} \) |
| 97 | \( 1 + 16.0iT - 97T^{2} \) |
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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
−8.619192644963770005072080640949, −7.88311073701138387507445489124, −7.23344301620950322265390120194, −6.71846989065904306548246503054, −5.99543272595935785218353106109, −4.95165761196791104020332783744, −3.93586855129452589246725951668, −3.26866806373276857884787222522, −2.60586581685535123931109887008, −1.64133950991910803610826615860,
0.20838370750067704535730500188, 1.57325676806904738414572944706, 2.23822957512158620994326775542, 3.47510861883497963815874656450, 4.40565769741137512605576050103, 4.82563526329731678716194748435, 5.70305245136188955715259454285, 6.61366038665795575862867332992, 7.35516613397102448347140560601, 8.354333859589197454650435816688
