| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s + 9-s − 6·11-s − 2·12-s + 3·13-s − 4·16-s + 4·17-s + 2·18-s + 19-s − 12·22-s + 4·23-s + 6·26-s − 27-s − 8·29-s + 31-s − 8·32-s + 6·33-s + 8·34-s + 2·36-s − 7·37-s + 2·38-s − 3·39-s − 6·41-s − 43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s − 1.80·11-s − 0.577·12-s + 0.832·13-s − 16-s + 0.970·17-s + 0.471·18-s + 0.229·19-s − 2.55·22-s + 0.834·23-s + 1.17·26-s − 0.192·27-s − 1.48·29-s + 0.179·31-s − 1.41·32-s + 1.04·33-s + 1.37·34-s + 1/3·36-s − 1.15·37-s + 0.324·38-s − 0.480·39-s − 0.937·41-s − 0.152·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−7.908723881980586150447988514319, −7.25443551019616964519259463431, −6.36270573651068181544479659187, −5.56137083626240592653082286606, −5.27799937003101454688566989739, −4.51570479554152650207693419765, −3.44947384790112759277428044139, −2.95938112484855431886799932997, −1.67437521076134331819095723511, 0,
1.67437521076134331819095723511, 2.95938112484855431886799932997, 3.44947384790112759277428044139, 4.51570479554152650207693419765, 5.27799937003101454688566989739, 5.56137083626240592653082286606, 6.36270573651068181544479659187, 7.25443551019616964519259463431, 7.908723881980586150447988514319
