L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s − 15-s − 4·19-s − 2·21-s + 25-s + 27-s − 6·29-s + 4·31-s + 2·35-s − 10·37-s − 12·41-s + 10·43-s − 45-s − 8·47-s − 3·49-s − 6·53-s − 4·57-s + 6·59-s − 6·61-s − 2·63-s − 6·67-s − 2·71-s − 12·73-s + 75-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.258·15-s − 0.917·19-s − 0.436·21-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.338·35-s − 1.64·37-s − 1.87·41-s + 1.52·43-s − 0.149·45-s − 1.16·47-s − 3/7·49-s − 0.824·53-s − 0.529·57-s + 0.781·59-s − 0.768·61-s − 0.251·63-s − 0.733·67-s − 0.237·71-s − 1.40·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8940393746\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8940393746\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 \) | |
good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−14.14938475892143, −13.61239557548013, −13.09241568679223, −12.81205229312434, −12.13127504688846, −11.79525891637262, −11.06932824039453, −10.54773263409190, −10.10795435894225, −9.535771200848667, −9.025659835524651, −8.497534720310333, −8.127737008769841, −7.339022652409386, −7.053551301983185, −6.327706753789146, −5.959168017740367, −5.009509487133355, −4.615385559275869, −3.780512377648843, −3.441964399945895, −2.857911765535312, −2.053298181564459, −1.461508394848185, −0.2928285994785624,
0.2928285994785624, 1.461508394848185, 2.053298181564459, 2.857911765535312, 3.441964399945895, 3.780512377648843, 4.615385559275869, 5.009509487133355, 5.959168017740367, 6.327706753789146, 7.053551301983185, 7.339022652409386, 8.127737008769841, 8.497534720310333, 9.025659835524651, 9.535771200848667, 10.10795435894225, 10.54773263409190, 11.06932824039453, 11.79525891637262, 12.13127504688846, 12.81205229312434, 13.09241568679223, 13.61239557548013, 14.14938475892143