| L(s) = 1 | + (−1.16 − 0.520i)2-s + (1.71 + 0.222i)3-s + (−0.241 − 0.267i)4-s + (2.12 − 0.708i)5-s + (−1.89 − 1.15i)6-s + (1.38 + 2.40i)7-s + (0.933 + 2.87i)8-s + (2.90 + 0.765i)9-s + (−2.85 − 0.275i)10-s + (−3.72 − 1.66i)11-s + (−0.354 − 0.513i)12-s + (1.61 − 0.717i)13-s + (−0.372 − 3.54i)14-s + (3.80 − 0.744i)15-s + (0.329 − 3.13i)16-s + (−1.70 − 5.25i)17-s + ⋯ |
| L(s) = 1 | + (−0.827 − 0.368i)2-s + (0.991 + 0.128i)3-s + (−0.120 − 0.133i)4-s + (0.948 − 0.316i)5-s + (−0.772 − 0.471i)6-s + (0.525 + 0.909i)7-s + (0.330 + 1.01i)8-s + (0.966 + 0.255i)9-s + (−0.901 − 0.0871i)10-s + (−1.12 − 0.500i)11-s + (−0.102 − 0.148i)12-s + (0.446 − 0.198i)13-s + (−0.0994 − 0.946i)14-s + (0.981 − 0.192i)15-s + (0.0823 − 0.783i)16-s + (−0.413 − 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18113 - 0.279514i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18113 - 0.279514i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.71 - 0.222i)T \) |
| 5 | \( 1 + (-2.12 + 0.708i)T \) |
| good | 2 | \( 1 + (1.16 + 0.520i)T + (1.33 + 1.48i)T^{2} \) |
| 7 | \( 1 + (-1.38 - 2.40i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.72 + 1.66i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-1.61 + 0.717i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (1.70 + 5.25i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.48 - 7.65i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.403 + 3.83i)T + (-22.4 + 4.78i)T^{2} \) |
| 29 | \( 1 + (-3.00 + 0.638i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (7.56 + 1.60i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (5.00 + 3.63i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (8.98 - 4.00i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-1.87 - 3.24i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.343 - 0.0729i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (2.83 - 8.73i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (10.7 - 4.79i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-3.54 - 1.57i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (6.62 + 1.40i)T + (61.2 + 27.2i)T^{2} \) |
| 71 | \( 1 + (1.46 - 4.49i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.59 + 1.15i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (3.83 - 0.815i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-8.05 + 8.94i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (2.09 - 1.52i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (0.0744 - 0.0158i)T + (88.6 - 39.4i)T^{2} \) |
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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
−12.14997141655984833714437288289, −10.76633985485818796529627634874, −10.06876658174063235687723006510, −9.155315513480084588903726179406, −8.555185811980436601272728283248, −7.74291578153202221840804478786, −5.76761811399472968583097222823, −4.91971260033966140436667602140, −2.79010186200985515943405628374, −1.70903095629744780193293930071,
1.72930665091513216704109175899, 3.44193885452011612713667764787, 4.86500247021352276106674748388, 6.79497843611253187756520963964, 7.42042149624940367643491528189, 8.427099960301565308998519212804, 9.229735500818459434623524760039, 10.17813621467104079040044316350, 10.81618408897945196140463653280, 12.76098554813192694081357281237
