| L(s) = 1 | − 3-s − 2·9-s + 6·11-s − 2·13-s + 6·17-s + 8·19-s − 3·23-s + 5·27-s + 3·29-s + 2·31-s − 6·33-s − 8·37-s + 2·39-s − 3·41-s − 5·43-s − 6·51-s − 12·53-s − 8·57-s − 61-s + 7·67-s + 3·69-s + 10·73-s − 4·79-s + 81-s − 3·83-s − 3·87-s − 3·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 1.80·11-s − 0.554·13-s + 1.45·17-s + 1.83·19-s − 0.625·23-s + 0.962·27-s + 0.557·29-s + 0.359·31-s − 1.04·33-s − 1.31·37-s + 0.320·39-s − 0.468·41-s − 0.762·43-s − 0.840·51-s − 1.64·53-s − 1.05·57-s − 0.128·61-s + 0.855·67-s + 0.361·69-s + 1.17·73-s − 0.450·79-s + 1/9·81-s − 0.329·83-s − 0.321·87-s − 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.713924759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.713924759\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−8.255068422866006050104579822444, −7.47412515521946972887964479492, −6.72955758821946826369657074289, −6.08726621013549838366168527268, −5.35906560962029520584328319073, −4.75436843540763875036298944584, −3.56939663785267587175216764450, −3.12993958273370302783212199139, −1.67317137806908499175125329209, −0.78409287570769326854101260467,
0.78409287570769326854101260467, 1.67317137806908499175125329209, 3.12993958273370302783212199139, 3.56939663785267587175216764450, 4.75436843540763875036298944584, 5.35906560962029520584328319073, 6.08726621013549838366168527268, 6.72955758821946826369657074289, 7.47412515521946972887964479492, 8.255068422866006050104579822444
