| L(s) = 1 | − 2·2-s + 2·4-s + 7-s + 6·11-s + 7·13-s − 2·14-s − 4·16-s − 4·17-s + 5·19-s − 12·22-s − 2·23-s − 14·26-s + 2·28-s + 8·29-s + 3·31-s + 8·32-s + 8·34-s − 37-s − 10·38-s − 2·41-s + 11·43-s + 12·44-s + 4·46-s − 4·47-s − 6·49-s + 14·52-s − 16·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.377·7-s + 1.80·11-s + 1.94·13-s − 0.534·14-s − 16-s − 0.970·17-s + 1.14·19-s − 2.55·22-s − 0.417·23-s − 2.74·26-s + 0.377·28-s + 1.48·29-s + 0.538·31-s + 1.41·32-s + 1.37·34-s − 0.164·37-s − 1.62·38-s − 0.312·41-s + 1.67·43-s + 1.80·44-s + 0.589·46-s − 0.583·47-s − 6/7·49-s + 1.94·52-s − 2.10·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.503886710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.503886710\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−8.168268623845037455913499448104, −7.19742145163329386987155244483, −6.49495034385590527907644595133, −6.21228954316337941361470699073, −4.97529838871611785628064012257, −4.12766449643564224535057912526, −3.54052573739090212445432495723, −2.27068121049293954646228585797, −1.27646316047069070839849832366, −0.930827996260469354147557631622,
0.930827996260469354147557631622, 1.27646316047069070839849832366, 2.27068121049293954646228585797, 3.54052573739090212445432495723, 4.12766449643564224535057912526, 4.97529838871611785628064012257, 6.21228954316337941361470699073, 6.49495034385590527907644595133, 7.19742145163329386987155244483, 8.168268623845037455913499448104
