| L(s) = 1 | − 0.778·3-s + 4.82·7-s − 2.39·9-s − 6.13·11-s − 1.15·13-s − 7.51·17-s + 3.38·19-s − 3.75·21-s + 2.19·23-s + 4.19·27-s + 29-s + 9.17·31-s + 4.77·33-s + 5.95·37-s + 0.901·39-s + 7.70·41-s + 2.84·43-s + 5.15·47-s + 16.3·49-s + 5.84·51-s − 8.39·53-s − 2.63·57-s − 8.52·59-s + 5.02·61-s − 11.5·63-s + 2.29·67-s − 1.70·69-s + ⋯ |
| L(s) = 1 | − 0.449·3-s + 1.82·7-s − 0.798·9-s − 1.85·11-s − 0.321·13-s − 1.82·17-s + 0.777·19-s − 0.819·21-s + 0.456·23-s + 0.807·27-s + 0.185·29-s + 1.64·31-s + 0.831·33-s + 0.979·37-s + 0.144·39-s + 1.20·41-s + 0.433·43-s + 0.752·47-s + 2.32·49-s + 0.818·51-s − 1.15·53-s − 0.349·57-s − 1.10·59-s + 0.642·61-s − 1.45·63-s + 0.279·67-s − 0.205·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.437543704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.437543704\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.778T + 3T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 + 6.13T + 11T^{2} \) |
| 13 | \( 1 + 1.15T + 13T^{2} \) |
| 17 | \( 1 + 7.51T + 17T^{2} \) |
| 19 | \( 1 - 3.38T + 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 31 | \( 1 - 9.17T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 - 7.70T + 41T^{2} \) |
| 43 | \( 1 - 2.84T + 43T^{2} \) |
| 47 | \( 1 - 5.15T + 47T^{2} \) |
| 53 | \( 1 + 8.39T + 53T^{2} \) |
| 59 | \( 1 + 8.52T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 - 2.29T + 67T^{2} \) |
| 71 | \( 1 - 9.81T + 71T^{2} \) |
| 73 | \( 1 - 0.495T + 73T^{2} \) |
| 79 | \( 1 - 8.19T + 79T^{2} \) |
| 83 | \( 1 - 1.10T + 83T^{2} \) |
| 89 | \( 1 - 4.82T + 89T^{2} \) |
| 97 | \( 1 + 3.41T + 97T^{2} \) |
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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
−8.556445606824709045993612493127, −7.996183210290389901424420942034, −7.50493140846625558792353553340, −6.38790205072187035484477623742, −5.49973589538446873840653743483, −4.88404887571550765592059868818, −4.46642465144997750235770361642, −2.76315067074540165724800979194, −2.25014932178571456027920362163, −0.73854598447220851296415488318,
0.73854598447220851296415488318, 2.25014932178571456027920362163, 2.76315067074540165724800979194, 4.46642465144997750235770361642, 4.88404887571550765592059868818, 5.49973589538446873840653743483, 6.38790205072187035484477623742, 7.50493140846625558792353553340, 7.996183210290389901424420942034, 8.556445606824709045993612493127
