| L(s) = 1 | + 0.0387·3-s + 5-s − 7-s − 2.99·9-s − 4.92·11-s + 1.04·13-s + 0.0387·15-s + 2.87·17-s − 2.41·19-s − 0.0387·21-s − 6.95·23-s + 25-s − 0.232·27-s + 3.44·29-s − 4.67·31-s − 0.190·33-s − 35-s + 9.21·37-s + 0.0405·39-s + 10.2·41-s + 43-s − 2.99·45-s − 5.98·47-s + 49-s + 0.111·51-s − 5.06·53-s − 4.92·55-s + ⋯ |
| L(s) = 1 | + 0.0223·3-s + 0.447·5-s − 0.377·7-s − 0.999·9-s − 1.48·11-s + 0.290·13-s + 0.0100·15-s + 0.697·17-s − 0.554·19-s − 0.00845·21-s − 1.45·23-s + 0.200·25-s − 0.0447·27-s + 0.639·29-s − 0.839·31-s − 0.0332·33-s − 0.169·35-s + 1.51·37-s + 0.00649·39-s + 1.59·41-s + 0.152·43-s − 0.446·45-s − 0.872·47-s + 0.142·49-s + 0.0156·51-s − 0.695·53-s − 0.663·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.312417866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.312417866\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 - 0.0387T + 3T^{2} \) |
| 11 | \( 1 + 4.92T + 11T^{2} \) |
| 13 | \( 1 - 1.04T + 13T^{2} \) |
| 17 | \( 1 - 2.87T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 23 | \( 1 + 6.95T + 23T^{2} \) |
| 29 | \( 1 - 3.44T + 29T^{2} \) |
| 31 | \( 1 + 4.67T + 31T^{2} \) |
| 37 | \( 1 - 9.21T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 47 | \( 1 + 5.98T + 47T^{2} \) |
| 53 | \( 1 + 5.06T + 53T^{2} \) |
| 59 | \( 1 - 8.66T + 59T^{2} \) |
| 61 | \( 1 - 4.21T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 0.593T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 - 6.82T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 2.99T + 97T^{2} \) |
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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
−7.890263483673318835099783211026, −7.70523640671552899931325784346, −6.39837514338771017887935077205, −5.95370113051797150525006211998, −5.38408368150846692516682592113, −4.51170066985336659553142056996, −3.49598166279969011201299306208, −2.71036768270083637795432602857, −2.08029422991873505290551525890, −0.56894505114254834563048223934,
0.56894505114254834563048223934, 2.08029422991873505290551525890, 2.71036768270083637795432602857, 3.49598166279969011201299306208, 4.51170066985336659553142056996, 5.38408368150846692516682592113, 5.95370113051797150525006211998, 6.39837514338771017887935077205, 7.70523640671552899931325784346, 7.890263483673318835099783211026
