| L(s) = 1 | + 0.573·3-s + 5.92·5-s − 31.1·7-s − 26.6·9-s − 19.0·11-s + 27.3·13-s + 3.40·15-s − 27.5·17-s + 19·19-s − 17.8·21-s − 162.·23-s − 89.8·25-s − 30.7·27-s − 66.9·29-s − 127.·31-s − 10.9·33-s − 184.·35-s + 273.·37-s + 15.7·39-s + 228.·41-s + 198.·43-s − 158.·45-s − 573.·47-s + 627.·49-s − 15.8·51-s + 610.·53-s − 113.·55-s + ⋯ |
| L(s) = 1 | + 0.110·3-s + 0.530·5-s − 1.68·7-s − 0.987·9-s − 0.522·11-s + 0.584·13-s + 0.0585·15-s − 0.393·17-s + 0.229·19-s − 0.185·21-s − 1.47·23-s − 0.718·25-s − 0.219·27-s − 0.428·29-s − 0.738·31-s − 0.0577·33-s − 0.891·35-s + 1.21·37-s + 0.0645·39-s + 0.871·41-s + 0.702·43-s − 0.523·45-s − 1.77·47-s + 1.82·49-s − 0.0434·51-s + 1.58·53-s − 0.277·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 0.573T + 27T^{2} \) |
| 5 | \( 1 - 5.92T + 125T^{2} \) |
| 7 | \( 1 + 31.1T + 343T^{2} \) |
| 11 | \( 1 + 19.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 27.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 27.5T + 4.91e3T^{2} \) |
| 23 | \( 1 + 162.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 66.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 127.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 273.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 228.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 198.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 573.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 610.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 202.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 774.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 589.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 392.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 7.44T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 209.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 705.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 94.8T + 9.12e5T^{2} \) |
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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
−12.13640879466326728324488425927, −10.92496104423201899069151922293, −9.826969497993471674138334590518, −9.126948323911907312045603352790, −7.83284154416716621669940915247, −6.32403607658669603939084912250, −5.72739331047730278791906025315, −3.72512288367368997303150475052, −2.48981361693998056777603098813, 0,
2.48981361693998056777603098813, 3.72512288367368997303150475052, 5.72739331047730278791906025315, 6.32403607658669603939084912250, 7.83284154416716621669940915247, 9.126948323911907312045603352790, 9.826969497993471674138334590518, 10.92496104423201899069151922293, 12.13640879466326728324488425927
