L(s) = 1 | − 1.40·3-s + 5-s + 3.27·7-s − 1.03·9-s + 3.84·11-s − 5.20·13-s − 1.40·15-s + 0.810·17-s − 4.07·19-s − 4.59·21-s + 6.38·23-s + 25-s + 5.65·27-s − 7.98·29-s − 4.89·31-s − 5.39·33-s + 3.27·35-s − 7.95·37-s + 7.29·39-s + 10.5·41-s − 7.64·43-s − 1.03·45-s − 6.83·47-s + 3.70·49-s − 1.13·51-s + 1.56·53-s + 3.84·55-s + ⋯ |
L(s) = 1 | − 0.810·3-s + 0.447·5-s + 1.23·7-s − 0.343·9-s + 1.15·11-s − 1.44·13-s − 0.362·15-s + 0.196·17-s − 0.935·19-s − 1.00·21-s + 1.33·23-s + 0.200·25-s + 1.08·27-s − 1.48·29-s − 0.879·31-s − 0.939·33-s + 0.553·35-s − 1.30·37-s + 1.16·39-s + 1.65·41-s − 1.16·43-s − 0.153·45-s − 0.996·47-s + 0.529·49-s − 0.159·51-s + 0.215·53-s + 0.518·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 + 1.40T + 3T^{2} \) |
| 7 | \( 1 - 3.27T + 7T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 + 5.20T + 13T^{2} \) |
| 17 | \( 1 - 0.810T + 17T^{2} \) |
| 19 | \( 1 + 4.07T + 19T^{2} \) |
| 23 | \( 1 - 6.38T + 23T^{2} \) |
| 29 | \( 1 + 7.98T + 29T^{2} \) |
| 31 | \( 1 + 4.89T + 31T^{2} \) |
| 37 | \( 1 + 7.95T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 7.64T + 43T^{2} \) |
| 47 | \( 1 + 6.83T + 47T^{2} \) |
| 53 | \( 1 - 1.56T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 7.68T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 8.04T + 73T^{2} \) |
| 79 | \( 1 + 5.13T + 79T^{2} \) |
| 83 | \( 1 - 0.0490T + 83T^{2} \) |
| 89 | \( 1 - 7.75T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.48949316140432001067272907691, −7.07286072358358605769535178921, −6.19650326569722856258621416994, −5.50321141621896381028435706043, −4.93565848074969366661441367961, −4.34627312984043954508529326164, −3.19732019457185512491033874461, −2.09415264406655240356120267044, −1.37667688092273586087692392041, 0,
1.37667688092273586087692392041, 2.09415264406655240356120267044, 3.19732019457185512491033874461, 4.34627312984043954508529326164, 4.93565848074969366661441367961, 5.50321141621896381028435706043, 6.19650326569722856258621416994, 7.07286072358358605769535178921, 7.48949316140432001067272907691