| L(s) = 1 | + 5-s − 2·11-s − 2·13-s − 4·17-s − 19-s + 23-s − 4·25-s − 9·29-s + 2·31-s + 4·43-s − 9·47-s − 7·49-s − 5·53-s − 2·55-s + 10·59-s + 4·61-s − 2·65-s + 11·67-s − 13·71-s + 73-s + 4·79-s − 16·83-s − 4·85-s − 95-s − 13·97-s − 9·101-s + 16·103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.603·11-s − 0.554·13-s − 0.970·17-s − 0.229·19-s + 0.208·23-s − 4/5·25-s − 1.67·29-s + 0.359·31-s + 0.609·43-s − 1.31·47-s − 49-s − 0.686·53-s − 0.269·55-s + 1.30·59-s + 0.512·61-s − 0.248·65-s + 1.34·67-s − 1.54·71-s + 0.117·73-s + 0.450·79-s − 1.75·83-s − 0.433·85-s − 0.102·95-s − 1.31·97-s − 0.895·101-s + 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−8.840696644641943506985303357027, −8.008426431526685611289104828253, −7.24361260352940556790341961199, −6.40624284722563559960582802525, −5.56384170137228753013296750920, −4.80078702476556849636707239968, −3.81820876741906552147322604159, −2.63766108773545464456423065215, −1.79833846435937955132229668939, 0,
1.79833846435937955132229668939, 2.63766108773545464456423065215, 3.81820876741906552147322604159, 4.80078702476556849636707239968, 5.56384170137228753013296750920, 6.40624284722563559960582802525, 7.24361260352940556790341961199, 8.008426431526685611289104828253, 8.840696644641943506985303357027
