L(s) = 1 | + (1.34 − 0.437i)2-s + (1.61 − 1.17i)4-s + (−3.10 − 4.27i)7-s + (1.66 − 2.28i)8-s + (0.927 + 2.85i)9-s + (3.30 + 0.217i)11-s + (3.42 − 1.11i)13-s + (−6.04 − 4.39i)14-s + (1.23 − 3.80i)16-s + (−5.73 − 1.86i)17-s + (2.49 + 3.43i)18-s + (1.60 − 2.20i)19-s + (4.54 − 1.15i)22-s + (−4.04 − 2.93i)25-s + (4.12 − 2.99i)26-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (−1.17 − 1.61i)7-s + (0.587 − 0.809i)8-s + (0.309 + 0.951i)9-s + (0.997 + 0.0656i)11-s + (0.951 − 0.309i)13-s + (−1.61 − 1.17i)14-s + (0.309 − 0.951i)16-s + (−1.39 − 0.452i)17-s + (0.587 + 0.809i)18-s + (0.367 − 0.505i)19-s + (0.969 − 0.245i)22-s + (−0.809 − 0.587i)25-s + (0.809 − 0.587i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.267 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.96589 - 1.49497i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.96589 - 1.49497i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.34 + 0.437i)T \) |
| 11 | \( 1 + (-3.30 - 0.217i)T \) |
| 13 | \( 1 + (-3.42 + 1.11i)T \) |
good | 3 | \( 1 + (-0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (3.10 + 4.27i)T + (-2.16 + 6.65i)T^{2} \) |
| 17 | \( 1 + (5.73 + 1.86i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.60 + 2.20i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (-6.19 - 8.52i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.523 + 1.61i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + (-8.51 - 6.18i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-3.11 - 9.60i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (10.6 - 7.76i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-7.74 - 2.51i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 + (0.617 - 1.90i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (16.1 + 5.25i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (78.4 - 57.0i)T^{2} \) |
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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
−10.69241222997866708891602706706, −10.04102437022390960390231600076, −8.978746039817371677279377454418, −7.41562836099327665654812884875, −6.84662871998369623145767724781, −6.04838596662943098611396584048, −4.54514725943303376405733888369, −3.99480474638545008966487567300, −2.85615679774785083395335852790, −1.15006971032326475047628089855,
2.08947911116533873574367593168, 3.40067371515296877251138985087, 4.10582474890221271061372143589, 5.66362894157888913166787216784, 6.35423909710787469669343718911, 6.72633826422255343574565242861, 8.400330427293186967588284037339, 9.048028431377130009419505092757, 9.905338231327492233529577655359, 11.41865023191250080315475334579