| L(s) = 1 | + 2-s + 4-s + 3·7-s + 8-s + 7·13-s + 3·14-s + 16-s + 3·17-s + 5·19-s + 8·23-s + 7·26-s + 3·28-s − 5·29-s − 8·31-s + 32-s + 3·34-s − 11·37-s + 5·38-s − 4·41-s − 6·43-s + 8·46-s + 4·47-s + 2·49-s + 7·52-s − 6·53-s + 3·56-s − 5·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.13·7-s + 0.353·8-s + 1.94·13-s + 0.801·14-s + 1/4·16-s + 0.727·17-s + 1.14·19-s + 1.66·23-s + 1.37·26-s + 0.566·28-s − 0.928·29-s − 1.43·31-s + 0.176·32-s + 0.514·34-s − 1.80·37-s + 0.811·38-s − 0.624·41-s − 0.914·43-s + 1.17·46-s + 0.583·47-s + 2/7·49-s + 0.970·52-s − 0.824·53-s + 0.400·56-s − 0.656·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.534000151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.534000151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.20636167011295, −14.05172157194100, −13.48370255715262, −12.85188565536214, −12.61717552488631, −11.66577194889130, −11.36399694046143, −11.10322832073730, −10.53059803749360, −9.866435274508734, −9.129572066935876, −8.605714705529447, −8.194633377886336, −7.509337006227618, −6.993224171838429, −6.513944802486596, −5.476371504788563, −5.412256502501600, −4.948957048495069, −3.855217873206684, −3.624025008109808, −3.068766327398207, −1.961987681203604, −1.476974211064255, −0.8530205590906021,
0.8530205590906021, 1.476974211064255, 1.961987681203604, 3.068766327398207, 3.624025008109808, 3.855217873206684, 4.948957048495069, 5.412256502501600, 5.476371504788563, 6.513944802486596, 6.993224171838429, 7.509337006227618, 8.194633377886336, 8.605714705529447, 9.129572066935876, 9.866435274508734, 10.53059803749360, 11.10322832073730, 11.36399694046143, 11.66577194889130, 12.61717552488631, 12.85188565536214, 13.48370255715262, 14.05172157194100, 14.20636167011295
