| L(s) = 1 | − 8·4-s + 37·7-s + 70·13-s + 64·16-s − 163·19-s − 296·28-s − 19·31-s + 433·37-s − 449·43-s + 1.02e3·49-s − 560·52-s + 719·61-s − 512·64-s + 880·67-s + 271·73-s + 1.30e3·76-s + 503·79-s + 2.59e3·91-s + 523·97-s + 19·103-s + 2.21e3·109-s + 2.36e3·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.99·7-s + 1.49·13-s + 16-s − 1.96·19-s − 1.99·28-s − 0.110·31-s + 1.92·37-s − 1.59·43-s + 2.99·49-s − 1.49·52-s + 1.50·61-s − 64-s + 1.60·67-s + 0.434·73-s + 1.96·76-s + 0.716·79-s + 2.98·91-s + 0.547·97-s + 0.0181·103-s + 1.94·109-s + 1.99·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.092191917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.092191917\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 - 37 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 70 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + 163 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 + 19 T + p^{3} T^{2} \) |
| 37 | \( 1 - 433 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 + 449 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 719 T + p^{3} T^{2} \) |
| 67 | \( 1 - 880 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 - 271 T + p^{3} T^{2} \) |
| 79 | \( 1 - 503 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 - 523 T + p^{3} T^{2} \) |
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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
−10.19709748093168646847124450563, −8.934996418073008366173741484205, −8.359192010532583017226896003646, −7.927020446947697271218676001671, −6.41788289944158183991936719370, −5.38507096834400289672157324812, −4.51725642100243525625514693965, −3.85703507400000164659837393614, −2.00935243896319121209988269546, −0.905007968081566446347284821587,
0.905007968081566446347284821587, 2.00935243896319121209988269546, 3.85703507400000164659837393614, 4.51725642100243525625514693965, 5.38507096834400289672157324812, 6.41788289944158183991936719370, 7.927020446947697271218676001671, 8.359192010532583017226896003646, 8.934996418073008366173741484205, 10.19709748093168646847124450563
