| L(s) = 1 | − 3-s − 5-s − 3·7-s + 9-s − 4.31·11-s − 2.31·13-s + 15-s − 3.31·17-s − 1.68·19-s + 3·21-s + 23-s + 25-s − 27-s − 0.683·29-s − 9.63·31-s + 4.31·33-s + 3·35-s + 11.6·37-s + 2.31·39-s + 3.31·41-s + 2.63·43-s − 45-s + 10.9·47-s + 2·49-s + 3.31·51-s − 9.94·53-s + 4.31·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.13·7-s + 0.333·9-s − 1.30·11-s − 0.642·13-s + 0.258·15-s − 0.804·17-s − 0.386·19-s + 0.654·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s − 0.126·29-s − 1.73·31-s + 0.751·33-s + 0.507·35-s + 1.91·37-s + 0.370·39-s + 0.517·41-s + 0.401·43-s − 0.149·45-s + 1.59·47-s + 0.285·49-s + 0.464·51-s − 1.36·53-s + 0.582·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4932281180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4932281180\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 4.31T + 11T^{2} \) |
| 13 | \( 1 + 2.31T + 13T^{2} \) |
| 17 | \( 1 + 3.31T + 17T^{2} \) |
| 19 | \( 1 + 1.68T + 19T^{2} \) |
| 29 | \( 1 + 0.683T + 29T^{2} \) |
| 31 | \( 1 + 9.63T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 3.31T + 41T^{2} \) |
| 43 | \( 1 - 2.63T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 9.94T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 3.68T + 61T^{2} \) |
| 67 | \( 1 - 2.36T + 67T^{2} \) |
| 71 | \( 1 + 7.94T + 71T^{2} \) |
| 73 | \( 1 - 1.68T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 5.31T + 83T^{2} \) |
| 89 | \( 1 - 8.63T + 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−9.010381023646908802259012859926, −7.77093506493199380898178001029, −7.40632280079315619107255902495, −6.45373924112261873036162725316, −5.81888135711848616296870184227, −4.91713868038794834125261151165, −4.14538393434009775991098574923, −3.08930521461905952475754796683, −2.24217080773876347982587974546, −0.41890730930205924386776626749,
0.41890730930205924386776626749, 2.24217080773876347982587974546, 3.08930521461905952475754796683, 4.14538393434009775991098574923, 4.91713868038794834125261151165, 5.81888135711848616296870184227, 6.45373924112261873036162725316, 7.40632280079315619107255902495, 7.77093506493199380898178001029, 9.010381023646908802259012859926
