| L(s) = 1 | + 20·3-s − 74·5-s − 24·7-s + 157·9-s + 124·11-s + 478·13-s − 1.48e3·15-s − 1.19e3·17-s + 3.04e3·19-s − 480·21-s + 184·23-s + 2.35e3·25-s − 1.72e3·27-s − 3.28e3·29-s − 5.72e3·31-s + 2.48e3·33-s + 1.77e3·35-s + 1.03e4·37-s + 9.56e3·39-s − 8.88e3·41-s − 9.18e3·43-s − 1.16e4·45-s + 2.36e4·47-s − 1.62e4·49-s − 2.39e4·51-s + 1.16e4·53-s − 9.17e3·55-s + ⋯ |
| L(s) = 1 | + 1.28·3-s − 1.32·5-s − 0.185·7-s + 0.646·9-s + 0.308·11-s + 0.784·13-s − 1.69·15-s − 1.00·17-s + 1.93·19-s − 0.237·21-s + 0.0725·23-s + 0.752·25-s − 0.454·27-s − 0.724·29-s − 1.07·31-s + 0.396·33-s + 0.245·35-s + 1.24·37-s + 1.00·39-s − 0.825·41-s − 0.757·43-s − 0.855·45-s + 1.56·47-s − 0.965·49-s − 1.28·51-s + 0.571·53-s − 0.409·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.289076427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.289076427\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 20 T + p^{5} T^{2} \) |
| 5 | \( 1 + 74 T + p^{5} T^{2} \) |
| 7 | \( 1 + 24 T + p^{5} T^{2} \) |
| 11 | \( 1 - 124 T + p^{5} T^{2} \) |
| 13 | \( 1 - 478 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1198 T + p^{5} T^{2} \) |
| 19 | \( 1 - 3044 T + p^{5} T^{2} \) |
| 23 | \( 1 - 8 p T + p^{5} T^{2} \) |
| 29 | \( 1 + 3282 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5728 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10326 T + p^{5} T^{2} \) |
| 41 | \( 1 + 8886 T + p^{5} T^{2} \) |
| 43 | \( 1 + 9188 T + p^{5} T^{2} \) |
| 47 | \( 1 - 23664 T + p^{5} T^{2} \) |
| 53 | \( 1 - 11686 T + p^{5} T^{2} \) |
| 59 | \( 1 - 16876 T + p^{5} T^{2} \) |
| 61 | \( 1 + 18482 T + p^{5} T^{2} \) |
| 67 | \( 1 + 15532 T + p^{5} T^{2} \) |
| 71 | \( 1 + 31960 T + p^{5} T^{2} \) |
| 73 | \( 1 + 4886 T + p^{5} T^{2} \) |
| 79 | \( 1 - 44560 T + p^{5} T^{2} \) |
| 83 | \( 1 - 67364 T + p^{5} T^{2} \) |
| 89 | \( 1 - 71994 T + p^{5} T^{2} \) |
| 97 | \( 1 - 48866 T + p^{5} T^{2} \) |
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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
−20.38272644330273207744974954926, −19.81556891015865889600886934465, −18.49318108316314250087606226415, −16.08419269894389301856818544898, −14.97964116072561748695841291743, −13.47591864616889720982954873651, −11.51473330391714330469604565410, −9.044045596108991432116545123050, −7.61071476793259109846129100307, −3.59039350304072868397022548436,
3.59039350304072868397022548436, 7.61071476793259109846129100307, 9.044045596108991432116545123050, 11.51473330391714330469604565410, 13.47591864616889720982954873651, 14.97964116072561748695841291743, 16.08419269894389301856818544898, 18.49318108316314250087606226415, 19.81556891015865889600886934465, 20.38272644330273207744974954926
