| L(s) = 1 | − 3-s − 2·5-s − 2·7-s + 9-s + 6·11-s − 13-s + 2·15-s − 2·17-s + 6·19-s + 2·21-s − 25-s − 27-s − 6·29-s − 6·31-s − 6·33-s + 4·35-s + 2·37-s + 39-s − 10·41-s − 8·43-s − 2·45-s − 6·47-s − 3·49-s + 2·51-s + 6·53-s − 12·55-s − 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 1.80·11-s − 0.277·13-s + 0.516·15-s − 0.485·17-s + 1.37·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.07·31-s − 1.04·33-s + 0.676·35-s + 0.328·37-s + 0.160·39-s − 1.56·41-s − 1.21·43-s − 0.298·45-s − 0.875·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s − 1.61·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 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
−9.434167209011293153562224190133, −8.557800849424999652983582883770, −7.40042573560376430524739468702, −6.87793060440785568895454535100, −6.05077845640649423202880694074, −5.01006909807833067576141602994, −3.90937860380759759980864189364, −3.38284846594751756938281755069, −1.54514640905416338743467358576, 0,
1.54514640905416338743467358576, 3.38284846594751756938281755069, 3.90937860380759759980864189364, 5.01006909807833067576141602994, 6.05077845640649423202880694074, 6.87793060440785568895454535100, 7.40042573560376430524739468702, 8.557800849424999652983582883770, 9.434167209011293153562224190133
