| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s + 4·11-s + 6·13-s − 4·14-s + 16-s − 6·17-s − 19-s − 4·22-s + 4·23-s − 6·26-s + 4·28-s − 6·29-s − 8·31-s − 32-s + 6·34-s − 2·37-s + 38-s − 10·41-s + 8·43-s + 4·44-s − 4·46-s + 12·47-s + 9·49-s + 6·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s + 1.20·11-s + 1.66·13-s − 1.06·14-s + 1/4·16-s − 1.45·17-s − 0.229·19-s − 0.852·22-s + 0.834·23-s − 1.17·26-s + 0.755·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s − 0.328·37-s + 0.162·38-s − 1.56·41-s + 1.21·43-s + 0.603·44-s − 0.589·46-s + 1.75·47-s + 9/7·49-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.148209810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.148209810\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 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
−8.013745548134849687039261237633, −7.00124206579927964275182297659, −6.69922149458188066572624286892, −5.74262956677626990062236264129, −5.10564444641027336549584546184, −4.02173378191983997310108924575, −3.72016441642838448550397349657, −2.22906802971440979412846691852, −1.66384562243540249845230946187, −0.858118939074564005626241568249,
0.858118939074564005626241568249, 1.66384562243540249845230946187, 2.22906802971440979412846691852, 3.72016441642838448550397349657, 4.02173378191983997310108924575, 5.10564444641027336549584546184, 5.74262956677626990062236264129, 6.69922149458188066572624286892, 7.00124206579927964275182297659, 8.013745548134849687039261237633
