| L(s) = 1 | − 7-s + 11-s − 3·13-s + 5·19-s − 4·23-s − 5·25-s + 4·29-s + 4·31-s − 37-s − 8·41-s + 8·47-s − 6·49-s − 4·53-s − 8·59-s + 7·61-s + 5·67-s − 11·73-s − 77-s + 79-s − 16·83-s + 3·91-s + 5·97-s − 4·101-s + 7·103-s − 8·107-s − 14·109-s − 16·113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.301·11-s − 0.832·13-s + 1.14·19-s − 0.834·23-s − 25-s + 0.742·29-s + 0.718·31-s − 0.164·37-s − 1.24·41-s + 1.16·47-s − 6/7·49-s − 0.549·53-s − 1.04·59-s + 0.896·61-s + 0.610·67-s − 1.28·73-s − 0.113·77-s + 0.112·79-s − 1.75·83-s + 0.314·91-s + 0.507·97-s − 0.398·101-s + 0.689·103-s − 0.773·107-s − 1.34·109-s − 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−7.899639659968801323865626491372, −7.22268301833111580428238929903, −6.50562716996827814662675391337, −5.75858655251657148672809316193, −4.98726598447939846802133300397, −4.16465439838096608536318500749, −3.30204980510099588781783652047, −2.47859497501192258415586130991, −1.37606116557633349976225967203, 0,
1.37606116557633349976225967203, 2.47859497501192258415586130991, 3.30204980510099588781783652047, 4.16465439838096608536318500749, 4.98726598447939846802133300397, 5.75858655251657148672809316193, 6.50562716996827814662675391337, 7.22268301833111580428238929903, 7.899639659968801323865626491372
