| L(s) = 1 | − 11-s + 4·13-s − 2·17-s + 2·19-s − 8·23-s + 4·29-s − 4·31-s + 2·37-s + 12·41-s − 8·43-s − 4·47-s − 7·49-s − 10·53-s − 4·59-s + 6·61-s − 12·67-s + 16·73-s + 6·79-s − 2·83-s − 18·89-s + 2·97-s + 8·103-s − 10·107-s + 6·109-s + 2·113-s + ⋯ |
| L(s) = 1 | − 0.301·11-s + 1.10·13-s − 0.485·17-s + 0.458·19-s − 1.66·23-s + 0.742·29-s − 0.718·31-s + 0.328·37-s + 1.87·41-s − 1.21·43-s − 0.583·47-s − 49-s − 1.37·53-s − 0.520·59-s + 0.768·61-s − 1.46·67-s + 1.87·73-s + 0.675·79-s − 0.219·83-s − 1.90·89-s + 0.203·97-s + 0.788·103-s − 0.966·107-s + 0.574·109-s + 0.188·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−7.41868364540608009502061887386, −6.43346762460105995922821286205, −6.13352281464570022134646047504, −5.29448413105420875346972231743, −4.50529983138587167450041621241, −3.80650204919796961788065033610, −3.07356750790786723802085648161, −2.12213403288029276053835207806, −1.26639845063344440843353339231, 0,
1.26639845063344440843353339231, 2.12213403288029276053835207806, 3.07356750790786723802085648161, 3.80650204919796961788065033610, 4.50529983138587167450041621241, 5.29448413105420875346972231743, 6.13352281464570022134646047504, 6.43346762460105995922821286205, 7.41868364540608009502061887386
