| L(s) = 1 | + 0.810·2-s − 2.88·3-s − 1.34·4-s − 2.33·6-s − 3.73·7-s − 2.70·8-s + 5.34·9-s + 0.363·11-s + 3.88·12-s − 1.90·13-s − 3.02·14-s + 0.492·16-s − 1.56·17-s + 4.32·18-s − 4.74·19-s + 10.7·21-s + 0.294·22-s + 0.210·23-s + 7.82·24-s − 1.54·26-s − 6.76·27-s + 5.01·28-s + 3.72·29-s + 2.95·31-s + 5.81·32-s − 1.05·33-s − 1.27·34-s + ⋯ |
| L(s) = 1 | + 0.572·2-s − 1.66·3-s − 0.671·4-s − 0.955·6-s − 1.41·7-s − 0.957·8-s + 1.78·9-s + 0.109·11-s + 1.12·12-s − 0.529·13-s − 0.808·14-s + 0.123·16-s − 0.380·17-s + 1.02·18-s − 1.08·19-s + 2.35·21-s + 0.0628·22-s + 0.0439·23-s + 1.59·24-s − 0.303·26-s − 1.30·27-s + 0.948·28-s + 0.691·29-s + 0.530·31-s + 1.02·32-s − 0.182·33-s − 0.217·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 0.810T + 2T^{2} \) |
| 3 | \( 1 + 2.88T + 3T^{2} \) |
| 7 | \( 1 + 3.73T + 7T^{2} \) |
| 11 | \( 1 - 0.363T + 11T^{2} \) |
| 13 | \( 1 + 1.90T + 13T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 19 | \( 1 + 4.74T + 19T^{2} \) |
| 23 | \( 1 - 0.210T + 23T^{2} \) |
| 29 | \( 1 - 3.72T + 29T^{2} \) |
| 31 | \( 1 - 2.95T + 31T^{2} \) |
| 37 | \( 1 - 2.19T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 5.79T + 43T^{2} \) |
| 47 | \( 1 - 3.70T + 47T^{2} \) |
| 53 | \( 1 - 2.34T + 53T^{2} \) |
| 59 | \( 1 + 5.13T + 59T^{2} \) |
| 61 | \( 1 - 3.62T + 61T^{2} \) |
| 67 | \( 1 - 1.38T + 67T^{2} \) |
| 71 | \( 1 + 0.880T + 71T^{2} \) |
| 73 | \( 1 + 6.39T + 73T^{2} \) |
| 79 | \( 1 + 0.286T + 79T^{2} \) |
| 83 | \( 1 + 6.19T + 83T^{2} \) |
| 89 | \( 1 + 2.54T + 89T^{2} \) |
| 97 | \( 1 - 8.23T + 97T^{2} \) |
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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
−7.39112250928261150538753754276, −6.62163649243148173034400973921, −6.13709761812171585547505095794, −5.73320886168758725120705491927, −4.71764032736388428570770466841, −4.40735677439203849046074523796, −3.47999297009747627191503940404, −2.48709707015794495536098120628, −0.799470710600584116266900826664, 0,
0.799470710600584116266900826664, 2.48709707015794495536098120628, 3.47999297009747627191503940404, 4.40735677439203849046074523796, 4.71764032736388428570770466841, 5.73320886168758725120705491927, 6.13709761812171585547505095794, 6.62163649243148173034400973921, 7.39112250928261150538753754276
