| L(s) = 1 | − 5-s − 4·7-s + 3·11-s − 12·13-s − 5·17-s − 19-s − 7·23-s + 5·25-s − 4·29-s + 5·31-s + 4·35-s − 3·37-s + 4·41-s − 8·43-s + 5·47-s + 9·49-s − 53-s − 3·55-s + 15·59-s + 5·61-s + 12·65-s + 9·67-s − 7·73-s − 12·77-s − 79-s − 24·83-s + 5·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 0.904·11-s − 3.32·13-s − 1.21·17-s − 0.229·19-s − 1.45·23-s + 25-s − 0.742·29-s + 0.898·31-s + 0.676·35-s − 0.493·37-s + 0.624·41-s − 1.21·43-s + 0.729·47-s + 9/7·49-s − 0.137·53-s − 0.404·55-s + 1.95·59-s + 0.640·61-s + 1.48·65-s + 1.09·67-s − 0.819·73-s − 1.36·77-s − 0.112·79-s − 2.63·83-s + 0.542·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4747469650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4747469650\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 5 T - 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.49549802431122192381217601787, −10.47411426907964889454806856143, −10.06016489954286374747617162633, −9.791607641016762117965858236127, −9.652635685990278623902713620372, −8.789028958898603649441060969282, −8.730102225013387808816326458930, −7.935896359213675516277942525339, −7.28969035610257481088852742156, −6.97354850770304209183758257008, −6.84338532292723554744468877781, −6.09122931317079614960196505957, −5.62970814769919942823310160566, −4.68001347949940271995915261146, −4.63796167885260613766022337859, −3.85090533475972350874847749862, −3.27339628949014245546500783653, −2.43408927175023092257506212239, −2.19344941765543884341902698282, −0.38567653674790494324807772002,
0.38567653674790494324807772002, 2.19344941765543884341902698282, 2.43408927175023092257506212239, 3.27339628949014245546500783653, 3.85090533475972350874847749862, 4.63796167885260613766022337859, 4.68001347949940271995915261146, 5.62970814769919942823310160566, 6.09122931317079614960196505957, 6.84338532292723554744468877781, 6.97354850770304209183758257008, 7.28969035610257481088852742156, 7.935896359213675516277942525339, 8.730102225013387808816326458930, 8.789028958898603649441060969282, 9.652635685990278623902713620372, 9.791607641016762117965858236127, 10.06016489954286374747617162633, 10.47411426907964889454806856143, 11.49549802431122192381217601787
