| L(s) = 1 | + 2·3-s − 7-s + 9-s + 5·11-s + 8·17-s − 2·19-s − 2·21-s + 7·23-s − 4·27-s + 3·29-s − 4·31-s + 10·33-s + 37-s − 2·41-s + 3·43-s − 6·47-s + 49-s + 16·51-s − 10·53-s − 4·57-s − 4·59-s + 6·61-s − 63-s + 13·67-s + 14·69-s − 5·71-s + 6·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.50·11-s + 1.94·17-s − 0.458·19-s − 0.436·21-s + 1.45·23-s − 0.769·27-s + 0.557·29-s − 0.718·31-s + 1.74·33-s + 0.164·37-s − 0.312·41-s + 0.457·43-s − 0.875·47-s + 1/7·49-s + 2.24·51-s − 1.37·53-s − 0.529·57-s − 0.520·59-s + 0.768·61-s − 0.125·63-s + 1.58·67-s + 1.68·69-s − 0.593·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.833714380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.833714380\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−16.55750203953322, −15.87967200493573, −15.00872039733354, −14.76187988294117, −14.23099106676548, −13.87868320215368, −13.03581667959559, −12.57026446531382, −11.95458892678114, −11.31533910726020, −10.61677646640248, −9.776947627574880, −9.354279524260680, −8.968648659468199, −8.162693180615852, −7.750616239971045, −6.888245203973237, −6.406821631368046, −5.545837305158480, −4.779407325717320, −3.696844017068363, −3.479393100749738, −2.721221814183625, −1.716916706017293, −0.9034336507727742,
0.9034336507727742, 1.716916706017293, 2.721221814183625, 3.479393100749738, 3.696844017068363, 4.779407325717320, 5.545837305158480, 6.406821631368046, 6.888245203973237, 7.750616239971045, 8.162693180615852, 8.968648659468199, 9.354279524260680, 9.776947627574880, 10.61677646640248, 11.31533910726020, 11.95458892678114, 12.57026446531382, 13.03581667959559, 13.87868320215368, 14.23099106676548, 14.76187988294117, 15.00872039733354, 15.87967200493573, 16.55750203953322
