| L(s) = 1 | − 1.56i·2-s − 0.438·4-s − 4.56i·7-s − 2.43i·8-s + 2.56·11-s + i·13-s − 7.12·14-s − 4.68·16-s − 2.56i·17-s − 3.12·19-s − 4i·22-s − 6.56i·23-s + 1.56·26-s + 1.99i·28-s − 1.12·29-s + ⋯ |
| L(s) = 1 | − 1.10i·2-s − 0.219·4-s − 1.72i·7-s − 0.862i·8-s + 0.772·11-s + 0.277i·13-s − 1.90·14-s − 1.17·16-s − 0.621i·17-s − 0.716·19-s − 0.852i·22-s − 1.36i·23-s + 0.306·26-s + 0.377i·28-s − 0.208·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.727292247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.727292247\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 2 | \( 1 + 1.56iT - 2T^{2} \) |
| 7 | \( 1 + 4.56iT - 7T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 17 | \( 1 + 2.56iT - 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + 6.56iT - 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 1.68iT - 37T^{2} \) |
| 41 | \( 1 + 0.561T + 41T^{2} \) |
| 43 | \( 1 - 5.12iT - 43T^{2} \) |
| 47 | \( 1 - 2.87iT - 47T^{2} \) |
| 53 | \( 1 + 7.68iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 5.68T + 61T^{2} \) |
| 67 | \( 1 - 13.3iT - 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 7.68T + 79T^{2} \) |
| 83 | \( 1 + 16.4iT - 83T^{2} \) |
| 89 | \( 1 - 1.68T + 89T^{2} \) |
| 97 | \( 1 + 11.9iT - 97T^{2} \) |
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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
−8.428474053813568385167186498841, −7.50797090446430142763958545500, −6.70169268079819452928258506182, −6.40630759908832779382094789400, −4.70138885877770536539598188081, −4.22486912221671974006574538213, −3.48500939126526816548951121997, −2.50936491346950758331541196951, −1.38175453805479972862930024276, −0.54342970061074038321036766984,
1.74959036685978740967320433124, 2.57499287530024891199950292993, 3.71095809048708001844004660012, 4.92981475752757559947787239089, 5.56897309007758295193014550877, 6.21114224045757105056633516331, 6.69367410887226993629266058039, 7.80432609011119056961178574399, 8.270333730942216059708852040383, 9.051668637533927345062869053023
