| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 4·7-s − 8-s + 9-s + 6·11-s − 2·12-s − 2·13-s + 4·14-s + 16-s + 17-s − 18-s + 4·19-s + 8·21-s − 6·22-s + 2·24-s − 5·25-s + 2·26-s + 4·27-s − 4·28-s + 4·31-s − 32-s − 12·33-s − 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.577·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s + 1.74·21-s − 1.27·22-s + 0.408·24-s − 25-s + 0.392·26-s + 0.769·27-s − 0.755·28-s + 0.718·31-s − 0.176·32-s − 2.08·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7390824981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7390824981\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.55302301086507, −14.17676274974372, −13.56115764926673, −12.78655334684759, −12.39176907270304, −11.86072175428825, −11.55833209257535, −11.13620026488347, −10.21116505194529, −9.914284326743568, −9.525150763423065, −9.022055547722850, −8.376113152191075, −7.530224916217663, −7.029815514545548, −6.441170052604737, −6.226233640976757, −5.651406540124528, −4.917432884246612, −4.108377636026865, −3.450770763824966, −2.920504864009248, −1.921477332875915, −1.026802508681650, −0.4503075653132020,
0.4503075653132020, 1.026802508681650, 1.921477332875915, 2.920504864009248, 3.450770763824966, 4.108377636026865, 4.917432884246612, 5.651406540124528, 6.226233640976757, 6.441170052604737, 7.029815514545548, 7.530224916217663, 8.376113152191075, 9.022055547722850, 9.525150763423065, 9.914284326743568, 10.21116505194529, 11.13620026488347, 11.55833209257535, 11.86072175428825, 12.39176907270304, 12.78655334684759, 13.56115764926673, 14.17676274974372, 14.55302301086507
