| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 3·7-s + 8-s − 2·9-s + 10-s − 12-s − 3·14-s − 15-s + 16-s + 8·17-s − 2·18-s + 8·19-s + 20-s + 3·21-s − 24-s + 25-s + 5·27-s − 3·28-s + 2·29-s − 30-s + 6·31-s + 32-s + 8·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.288·12-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 1.94·17-s − 0.471·18-s + 1.83·19-s + 0.223·20-s + 0.654·21-s − 0.204·24-s + 1/5·25-s + 0.962·27-s − 0.566·28-s + 0.371·29-s − 0.182·30-s + 1.07·31-s + 0.176·32-s + 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.053604811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.053604811\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−9.964220279088027040372713572210, −9.117504034208809263091522087062, −7.88710686743086008344543156423, −7.05850740984927580765036901181, −5.99845531373657801488383612111, −5.73868658193238489196514418842, −4.79610477271126136919010002147, −3.33150020949949765199474388305, −2.89768926763918826004237282914, −1.03169730266403144045507001330,
1.03169730266403144045507001330, 2.89768926763918826004237282914, 3.33150020949949765199474388305, 4.79610477271126136919010002147, 5.73868658193238489196514418842, 5.99845531373657801488383612111, 7.05850740984927580765036901181, 7.88710686743086008344543156423, 9.117504034208809263091522087062, 9.964220279088027040372713572210
