| L(s) = 1 | − 3-s − 5-s + 9-s − 6·11-s + 13-s + 15-s + 3·17-s + 7·19-s + 4·23-s + 25-s − 27-s + 2·29-s − 4·31-s + 6·33-s − 4·37-s − 39-s + 5·41-s + 43-s − 45-s − 2·47-s − 3·51-s − 2·53-s + 6·55-s − 7·57-s + 2·59-s − 65-s + 6·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.80·11-s + 0.277·13-s + 0.258·15-s + 0.727·17-s + 1.60·19-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 1.04·33-s − 0.657·37-s − 0.160·39-s + 0.780·41-s + 0.152·43-s − 0.149·45-s − 0.291·47-s − 0.420·51-s − 0.274·53-s + 0.809·55-s − 0.927·57-s + 0.260·59-s − 0.124·65-s + 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 305760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 305760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−12.74596101027111, −12.57781681733339, −11.92093037316776, −11.46352776103125, −11.16095194409991, −10.62037362314326, −10.19316863153156, −9.893470026149412, −9.226538556619146, −8.755469536693190, −8.147074308702770, −7.728635890938011, −7.268584689538897, −7.104416824948064, −6.190919998885114, −5.738489878514160, −5.289942671634653, −4.977735199968196, −4.451507583989870, −3.665890240777870, −3.168273152742191, −2.824756938707216, −2.047598828484070, −1.251727079061526, −0.7212324801485619, 0,
0.7212324801485619, 1.251727079061526, 2.047598828484070, 2.824756938707216, 3.168273152742191, 3.665890240777870, 4.451507583989870, 4.977735199968196, 5.289942671634653, 5.738489878514160, 6.190919998885114, 7.104416824948064, 7.268584689538897, 7.728635890938011, 8.147074308702770, 8.755469536693190, 9.226538556619146, 9.893470026149412, 10.19316863153156, 10.62037362314326, 11.16095194409991, 11.46352776103125, 11.92093037316776, 12.57781681733339, 12.74596101027111
