L(s) = 1 | + (0.991 + 0.127i)3-s + (0.871 − 0.490i)4-s + (−0.462 − 0.886i)7-s + (0.967 + 0.253i)9-s + (0.926 − 0.375i)12-s + (−0.678 + 1.53i)13-s + (0.518 − 0.855i)16-s − 1.96·19-s + (−0.345 − 0.938i)21-s + (0.404 + 0.914i)25-s + (0.926 + 0.375i)27-s + (−0.838 − 0.545i)28-s + (−0.387 − 1.69i)31-s + (0.967 − 0.253i)36-s + (−0.318 − 0.0204i)37-s + ⋯ |
L(s) = 1 | + (0.991 + 0.127i)3-s + (0.871 − 0.490i)4-s + (−0.462 − 0.886i)7-s + (0.967 + 0.253i)9-s + (0.926 − 0.375i)12-s + (−0.678 + 1.53i)13-s + (0.518 − 0.855i)16-s − 1.96·19-s + (−0.345 − 0.938i)21-s + (0.404 + 0.914i)25-s + (0.926 + 0.375i)27-s + (−0.838 − 0.545i)28-s + (−0.387 − 1.69i)31-s + (0.967 − 0.253i)36-s + (−0.318 − 0.0204i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.571205487\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.571205487\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.991 - 0.127i)T \) |
| 7 | \( 1 + (0.462 + 0.886i)T \) |
good | 2 | \( 1 + (-0.871 + 0.490i)T^{2} \) |
| 5 | \( 1 + (-0.404 - 0.914i)T^{2} \) |
| 11 | \( 1 + (0.997 - 0.0640i)T^{2} \) |
| 13 | \( 1 + (0.678 - 1.53i)T + (-0.672 - 0.740i)T^{2} \) |
| 17 | \( 1 + (-0.991 - 0.127i)T^{2} \) |
| 19 | \( 1 + 1.96T + T^{2} \) |
| 23 | \( 1 + (0.345 + 0.938i)T^{2} \) |
| 29 | \( 1 + (0.345 - 0.938i)T^{2} \) |
| 31 | \( 1 + (0.387 + 1.69i)T + (-0.900 + 0.433i)T^{2} \) |
| 37 | \( 1 + (0.318 + 0.0204i)T + (0.991 + 0.127i)T^{2} \) |
| 41 | \( 1 + (0.981 - 0.191i)T^{2} \) |
| 43 | \( 1 + (-0.0460 - 0.0445i)T + (0.0320 + 0.999i)T^{2} \) |
| 47 | \( 1 + (-0.159 + 0.987i)T^{2} \) |
| 53 | \( 1 + (0.838 - 0.545i)T^{2} \) |
| 59 | \( 1 + (0.981 + 0.191i)T^{2} \) |
| 61 | \( 1 + (0.382 - 0.421i)T + (-0.0960 - 0.995i)T^{2} \) |
| 67 | \( 1 + (0.441 - 1.93i)T + (-0.900 - 0.433i)T^{2} \) |
| 71 | \( 1 + (0.345 + 0.938i)T^{2} \) |
| 73 | \( 1 + (-1.51 - 1.29i)T + (0.159 + 0.987i)T^{2} \) |
| 79 | \( 1 + (0.838 + 1.05i)T + (-0.222 + 0.974i)T^{2} \) |
| 83 | \( 1 + (0.572 + 0.820i)T^{2} \) |
| 89 | \( 1 + (-0.284 + 0.958i)T^{2} \) |
| 97 | \( 1 + (0.356 + 1.56i)T + (-0.900 + 0.433i)T^{2} \) |
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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
−9.992571670900321601363101173289, −9.419066360033732395563288923480, −8.502435274124111496313529050068, −7.32434869498527262614011614884, −6.99293974978080426062823387661, −6.08468895860829206209560313781, −4.57740148448832279134837701782, −3.86747159458894856949827795827, −2.57610285952999954784985384284, −1.72611657687410586405059253153,
2.05183761321917210076576245250, 2.79015943428700923209479444921, 3.55268908777920400166271146055, 4.93005598966655705763942087624, 6.21653835319105097637513572375, 6.83968225465693590779545057593, 7.889549520183340838136019249204, 8.398084351975119087393417737972, 9.153032500951053119146975549028, 10.33016195714492256420320356548