| L(s) = 1 | + 3-s − 7-s − 2·9-s − 4·13-s − 6·17-s + 2·19-s − 21-s − 23-s − 5·27-s − 2·29-s − 31-s + 9·37-s − 4·39-s − 6·41-s + 8·43-s + 8·47-s + 49-s − 6·51-s − 10·53-s + 2·57-s + 59-s + 2·61-s + 2·63-s − 11·67-s − 69-s + 11·71-s − 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s − 1.10·13-s − 1.45·17-s + 0.458·19-s − 0.218·21-s − 0.208·23-s − 0.962·27-s − 0.371·29-s − 0.179·31-s + 1.47·37-s − 0.640·39-s − 0.937·41-s + 1.21·43-s + 1.16·47-s + 1/7·49-s − 0.840·51-s − 1.37·53-s + 0.264·57-s + 0.130·59-s + 0.256·61-s + 0.251·63-s − 1.34·67-s − 0.120·69-s + 1.30·71-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 11 T + p T^{2} \) | 1.71.al |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.23405237549648, −13.66803343475032, −13.24372323253996, −12.83278323213296, −12.15075237226933, −11.77313705766078, −11.19579216727975, −10.73496499342650, −10.15388204240516, −9.417084893438597, −9.265769553627466, −8.776482478911126, −8.068557250931011, −7.610328470076394, −7.191275036992723, −6.447490883757566, −6.024160489905623, −5.395693536386512, −4.706623965185131, −4.266733819606703, −3.491241482322850, −2.961512433553683, −2.312901792470696, −1.994310311439380, −0.7574929506250348, 0,
0.7574929506250348, 1.994310311439380, 2.312901792470696, 2.961512433553683, 3.491241482322850, 4.266733819606703, 4.706623965185131, 5.395693536386512, 6.024160489905623, 6.447490883757566, 7.191275036992723, 7.610328470076394, 8.068557250931011, 8.776482478911126, 9.265769553627466, 9.417084893438597, 10.15388204240516, 10.73496499342650, 11.19579216727975, 11.77313705766078, 12.15075237226933, 12.83278323213296, 13.24372323253996, 13.66803343475032, 14.23405237549648
