| L(s) = 1 | + 7-s + 7·13-s − 7·19-s + 11·31-s + 10·37-s + 13·43-s − 6·49-s − 61-s − 11·67-s + 10·73-s − 4·79-s + 7·91-s + 19·97-s − 20·103-s + 17·109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.94·13-s − 1.60·19-s + 1.97·31-s + 1.64·37-s + 1.98·43-s − 6/7·49-s − 0.128·61-s − 1.34·67-s + 1.17·73-s − 0.450·79-s + 0.733·91-s + 1.92·97-s − 1.97·103-s + 1.62·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.746374314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.746374314\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 13 T + p T^{2} \) | 1.43.an |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−10.25755092224355481914678162833, −9.118748321594516525124036442583, −8.430865511782959865213729501526, −7.77626766652559721050033528508, −6.38941799487849744389628645812, −6.05116072640550684180077744012, −4.62797018585887042860441223425, −3.88604970335075171889592435415, −2.55410828099147883677623007727, −1.15127637725344994484767740360,
1.15127637725344994484767740360, 2.55410828099147883677623007727, 3.88604970335075171889592435415, 4.62797018585887042860441223425, 6.05116072640550684180077744012, 6.38941799487849744389628645812, 7.77626766652559721050033528508, 8.430865511782959865213729501526, 9.118748321594516525124036442583, 10.25755092224355481914678162833
