| L(s) = 1 | + 5-s − 4·7-s + 6·17-s − 4·23-s + 25-s + 2·29-s − 8·31-s − 4·35-s − 6·37-s − 6·41-s − 12·43-s + 12·47-s + 9·49-s + 10·53-s − 8·59-s − 10·61-s − 12·67-s − 8·71-s − 10·73-s − 16·79-s − 12·83-s + 6·85-s − 6·89-s − 18·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s + 1.45·17-s − 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s − 0.676·35-s − 0.986·37-s − 0.937·41-s − 1.82·43-s + 1.75·47-s + 9/7·49-s + 1.37·53-s − 1.04·59-s − 1.28·61-s − 1.46·67-s − 0.949·71-s − 1.17·73-s − 1.80·79-s − 1.31·83-s + 0.650·85-s − 0.635·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.29703832035409, −12.90644532440570, −12.38081313808016, −12.00223862288957, −11.76798651865651, −10.81387706763846, −10.43458880968062, −10.00213672821188, −9.821849959049247, −9.160374214302738, −8.745750740811266, −8.325755603838772, −7.452525709596509, −7.240469375312488, −6.744126112340844, −6.049485289092188, −5.765742507583347, −5.409648778999220, −4.645743294507520, −3.933848100305909, −3.539179303149293, −2.963177449250945, −2.629500503727768, −1.584435328508014, −1.351685074155218, 0, 0,
1.351685074155218, 1.584435328508014, 2.629500503727768, 2.963177449250945, 3.539179303149293, 3.933848100305909, 4.645743294507520, 5.409648778999220, 5.765742507583347, 6.049485289092188, 6.744126112340844, 7.240469375312488, 7.452525709596509, 8.325755603838772, 8.745750740811266, 9.160374214302738, 9.821849959049247, 10.00213672821188, 10.43458880968062, 10.81387706763846, 11.76798651865651, 12.00223862288957, 12.38081313808016, 12.90644532440570, 13.29703832035409
