| L(s) = 1 | + 3-s + 2·7-s + 9-s − 2·11-s + 13-s + 2·17-s − 2·19-s + 2·21-s + 4·23-s + 27-s + 2·29-s − 2·31-s − 2·33-s + 6·37-s + 39-s − 2·41-s + 8·43-s + 6·47-s − 3·49-s + 2·51-s − 6·53-s − 2·57-s + 6·59-s − 2·61-s + 2·63-s + 14·67-s + 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.485·17-s − 0.458·19-s + 0.436·21-s + 0.834·23-s + 0.192·27-s + 0.371·29-s − 0.359·31-s − 0.348·33-s + 0.986·37-s + 0.160·39-s − 0.312·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s + 0.280·51-s − 0.824·53-s − 0.264·57-s + 0.781·59-s − 0.256·61-s + 0.251·63-s + 1.71·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.680218023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.680218023\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−8.361620942879244566151439349995, −7.86603417237536598581475961608, −7.19495974433674066658361148016, −6.28300645673977111494459626224, −5.38400030252674501352957208596, −4.68704813371568514323907102765, −3.84348059849834106290921608314, −2.89445607858070880475637568329, −2.07309259184517387372889348845, −0.954263316726802443781529583449,
0.954263316726802443781529583449, 2.07309259184517387372889348845, 2.89445607858070880475637568329, 3.84348059849834106290921608314, 4.68704813371568514323907102765, 5.38400030252674501352957208596, 6.28300645673977111494459626224, 7.19495974433674066658361148016, 7.86603417237536598581475961608, 8.361620942879244566151439349995
