| L(s) = 1 | + 7-s − 3·9-s − 2·11-s − 17-s + 6·19-s + 8·37-s + 6·41-s − 12·43-s − 4·47-s + 49-s + 6·53-s − 6·59-s − 2·61-s − 3·63-s − 8·67-s + 6·73-s − 2·77-s + 9·81-s − 6·83-s − 6·89-s + 2·97-s + 6·99-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s − 0.603·11-s − 0.242·17-s + 1.37·19-s + 1.31·37-s + 0.937·41-s − 1.82·43-s − 0.583·47-s + 1/7·49-s + 0.824·53-s − 0.781·59-s − 0.256·61-s − 0.377·63-s − 0.977·67-s + 0.702·73-s − 0.227·77-s + 81-s − 0.658·83-s − 0.635·89-s + 0.203·97-s + 0.603·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.90973683856388, −14.25770251066893, −13.84937766552049, −13.35466455141485, −12.87870214441432, −12.12695073519960, −11.69818646270651, −11.27905988630530, −10.80361195076023, −10.15747780292828, −9.583617700282766, −9.116655215395745, −8.434606980973533, −8.006007103486963, −7.529762460230539, −6.888954226471196, −6.139859465343153, −5.666055670227400, −5.103039480550240, −4.607947406403281, −3.776416267948153, −3.030278940978963, −2.656010986037446, −1.780713923562561, −0.9286558041607949, 0,
0.9286558041607949, 1.780713923562561, 2.656010986037446, 3.030278940978963, 3.776416267948153, 4.607947406403281, 5.103039480550240, 5.666055670227400, 6.139859465343153, 6.888954226471196, 7.529762460230539, 8.006007103486963, 8.434606980973533, 9.116655215395745, 9.583617700282766, 10.15747780292828, 10.80361195076023, 11.27905988630530, 11.69818646270651, 12.12695073519960, 12.87870214441432, 13.35466455141485, 13.84937766552049, 14.25770251066893, 14.90973683856388
