| L(s) = 1 | + 3-s + 5-s + 4.69i·7-s + 9-s − 1.75i·11-s + 4.69i·13-s + 15-s − 4.86·17-s + (−3.21 − 2.93i)19-s + 4.69i·21-s + 4.44i·23-s + 25-s + 27-s + 7.14i·29-s + 6.21·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.77i·7-s + 0.333·9-s − 0.528i·11-s + 1.30i·13-s + 0.258·15-s − 1.17·17-s + (−0.738 − 0.674i)19-s + 1.02i·21-s + 0.927i·23-s + 0.200·25-s + 0.192·27-s + 1.32i·29-s + 1.11·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 - 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.750278021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.750278021\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + (3.21 + 2.93i)T \) |
| good | 7 | \( 1 - 4.69iT - 7T^{2} \) |
| 11 | \( 1 + 1.75iT - 11T^{2} \) |
| 13 | \( 1 - 4.69iT - 13T^{2} \) |
| 17 | \( 1 + 4.86T + 17T^{2} \) |
| 23 | \( 1 - 4.44iT - 23T^{2} \) |
| 29 | \( 1 - 7.14iT - 29T^{2} \) |
| 31 | \( 1 - 6.21T + 31T^{2} \) |
| 37 | \( 1 + 6.20iT - 37T^{2} \) |
| 41 | \( 1 + 4.76iT - 41T^{2} \) |
| 43 | \( 1 + 6.20iT - 43T^{2} \) |
| 47 | \( 1 - 7.46iT - 47T^{2} \) |
| 53 | \( 1 - 4.44iT - 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 9.79T + 73T^{2} \) |
| 79 | \( 1 + 1.35T + 79T^{2} \) |
| 83 | \( 1 - 10.8iT - 83T^{2} \) |
| 89 | \( 1 - 15.5iT - 89T^{2} \) |
| 97 | \( 1 - 1.67iT - 97T^{2} \) |
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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
−8.810569881008000487831901529091, −8.145214480922468621313800424959, −7.00782899378138488167873414854, −6.44698413713632255453921947787, −5.72348459695912384377189709745, −4.93446594091518043708701445385, −4.11966391760457706667860301044, −2.97425888457489037795315764132, −2.33078958863853214793551429698, −1.65144556158986412850285318518,
0.40672576343319535643677622047, 1.53854868221792931938840250557, 2.57528048158978341751358870108, 3.43906416586518157199255090182, 4.47702916262619008042661091707, 4.63677897449071317424722089785, 6.14403602131438853898132731067, 6.58286810865698844539092934679, 7.43256985774748873727807329982, 8.074792641729398500793876197770
