| L(s) = 1 | − 2.82·3-s + 2.82·5-s + 5.00·9-s + 4·11-s − 2.82·13-s − 8.00·15-s − 5.65·17-s + 2.82·19-s + 3.00·25-s − 5.65·27-s + 2·29-s + 5.65·31-s − 11.3·33-s + 10·37-s + 8.00·39-s + 5.65·41-s + 4·43-s + 14.1·45-s − 5.65·47-s + 16.0·51-s + 6·53-s + 11.3·55-s − 8.00·57-s + 2.82·59-s + 14.1·61-s − 8.00·65-s − 12·67-s + ⋯ |
| L(s) = 1 | − 1.63·3-s + 1.26·5-s + 1.66·9-s + 1.20·11-s − 0.784·13-s − 2.06·15-s − 1.37·17-s + 0.648·19-s + 0.600·25-s − 1.08·27-s + 0.371·29-s + 1.01·31-s − 1.96·33-s + 1.64·37-s + 1.28·39-s + 0.883·41-s + 0.609·43-s + 2.10·45-s − 0.825·47-s + 2.24·51-s + 0.824·53-s + 1.52·55-s − 1.05·57-s + 0.368·59-s + 1.81·61-s − 0.992·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.143122167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.143122167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 5.65T + 97T^{2} \) |
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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
−10.23507420689339436190875550945, −9.714211439987077746275688683316, −8.893897705432050623915662987054, −7.32970383997053312115234417078, −6.41351849838639183347756969123, −6.07146340232479521253817627894, −5.04869630676600092966456186248, −4.28676224987771675110877287235, −2.35387932585599464505735885145, −0.999351453326497234930672369111,
0.999351453326497234930672369111, 2.35387932585599464505735885145, 4.28676224987771675110877287235, 5.04869630676600092966456186248, 6.07146340232479521253817627894, 6.41351849838639183347756969123, 7.32970383997053312115234417078, 8.893897705432050623915662987054, 9.714211439987077746275688683316, 10.23507420689339436190875550945
