| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 3·11-s − 13-s + 15-s − 3·17-s − 6·19-s − 21-s − 23-s + 25-s − 27-s + 8·29-s + 4·31-s − 3·33-s − 35-s + 5·37-s + 39-s − 5·41-s + 6·43-s − 45-s − 8·47-s − 6·49-s + 3·51-s + 9·53-s − 3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 0.258·15-s − 0.727·17-s − 1.37·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.48·29-s + 0.718·31-s − 0.522·33-s − 0.169·35-s + 0.821·37-s + 0.160·39-s − 0.780·41-s + 0.914·43-s − 0.149·45-s − 1.16·47-s − 6/7·49-s + 0.420·51-s + 1.23·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.348887234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.348887234\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−8.147392979087412721172777571288, −7.23458504479582604575646854092, −6.47325104698522766506597591619, −6.20458800132620540339314942714, −4.98204046895154126305642654414, −4.48386660038448688215945832167, −3.87288093191979679201119230172, −2.71713837768019973587400504649, −1.74414017259913515534309893024, −0.63117973648377647551504246168,
0.63117973648377647551504246168, 1.74414017259913515534309893024, 2.71713837768019973587400504649, 3.87288093191979679201119230172, 4.48386660038448688215945832167, 4.98204046895154126305642654414, 6.20458800132620540339314942714, 6.47325104698522766506597591619, 7.23458504479582604575646854092, 8.147392979087412721172777571288
