| L(s) = 1 | + 3-s + 2·5-s + 2·7-s + 9-s + 13-s + 2·15-s + 17-s − 19-s + 2·21-s − 8·23-s − 25-s + 27-s + 6·29-s + 4·31-s + 4·35-s + 2·37-s + 39-s + 4·41-s − 7·43-s + 2·45-s − 13·47-s − 3·49-s + 51-s − 6·53-s − 57-s + 4·59-s + 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.516·15-s + 0.242·17-s − 0.229·19-s + 0.436·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.676·35-s + 0.328·37-s + 0.160·39-s + 0.624·41-s − 1.06·43-s + 0.298·45-s − 1.89·47-s − 3/7·49-s + 0.140·51-s − 0.824·53-s − 0.132·57-s + 0.520·59-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 17 T + p T^{2} \) | 1.83.r |
| 89 | \( 1 + 17 T + p T^{2} \) | 1.89.r |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.05831010259366, −13.52609739167494, −13.29150442113234, −12.52464587741537, −12.16729740546286, −11.43418714614970, −11.22179720865699, −10.32258993349651, −9.959516714233188, −9.782097585149815, −9.004586290199142, −8.439588228283798, −8.075829177461254, −7.729705203155392, −6.901112851045684, −6.243659459660279, −6.078352452333169, −5.234402855433498, −4.738513839057850, −4.211822370064992, −3.524986394002590, −2.859593317354498, −2.225420501519881, −1.681420858008620, −1.193668223383443, 0,
1.193668223383443, 1.681420858008620, 2.225420501519881, 2.859593317354498, 3.524986394002590, 4.211822370064992, 4.738513839057850, 5.234402855433498, 6.078352452333169, 6.243659459660279, 6.901112851045684, 7.729705203155392, 8.075829177461254, 8.439588228283798, 9.004586290199142, 9.782097585149815, 9.959516714233188, 10.32258993349651, 11.22179720865699, 11.43418714614970, 12.16729740546286, 12.52464587741537, 13.29150442113234, 13.52609739167494, 14.05831010259366
