L(s) = 1 | + (−0.366 + 1.36i)2-s − 1.73i·3-s + (−1.73 − i)4-s + (2.36 + 0.633i)6-s + (2 + 1.73i)7-s + (2 − 1.99i)8-s + 3.73i·11-s + (−1.73 + 2.99i)12-s − 6.46·13-s + (−3.09 + 2.09i)14-s + (1.99 + 3.46i)16-s + 0.464·17-s + 6·19-s + (2.99 − 3.46i)21-s + (−5.09 − 1.36i)22-s + 5.46·23-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s − 0.999i·3-s + (−0.866 − 0.5i)4-s + (0.965 + 0.258i)6-s + (0.755 + 0.654i)7-s + (0.707 − 0.707i)8-s + 1.12i·11-s + (−0.499 + 0.866i)12-s − 1.79·13-s + (−0.827 + 0.560i)14-s + (0.499 + 0.866i)16-s + 0.112·17-s + 1.37·19-s + (0.654 − 0.755i)21-s + (−1.08 − 0.291i)22-s + 1.13·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.21531 + 0.511764i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21531 + 0.511764i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.366 - 1.36i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 3 | \( 1 + 1.73iT - 3T^{2} \) |
| 11 | \( 1 - 3.73iT - 11T^{2} \) |
| 13 | \( 1 + 6.46T + 13T^{2} \) |
| 17 | \( 1 - 0.464T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 5.46T + 23T^{2} \) |
| 29 | \( 1 - 5.92T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 2.53iT - 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 1.73iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 2.53iT - 61T^{2} \) |
| 67 | \( 1 - 3.46T + 67T^{2} \) |
| 71 | \( 1 - 0.535iT - 71T^{2} \) |
| 73 | \( 1 - 0.928T + 73T^{2} \) |
| 79 | \( 1 - 2.66iT - 79T^{2} \) |
| 83 | \( 1 + 8.53iT - 83T^{2} \) |
| 89 | \( 1 + 9.46iT - 89T^{2} \) |
| 97 | \( 1 - 7.39T + 97T^{2} \) |
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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
−10.13591958560407003844042272150, −9.647247263125984775046465464776, −8.574940005459354093573387562060, −7.59277623670557322669348218525, −7.30466072634823926123553192529, −6.40556962816466258271657371312, −5.07432173052229602169971751052, −4.71176723060455395299061440156, −2.54651521480377421887113424480, −1.23652184555140662295150178703,
0.967151305491853518359811179307, 2.72907840047777876539199614572, 3.69125717956506998694355155800, 4.79524933030583867781703841285, 5.17298314825721475329119444735, 7.12142173936028115196088854512, 7.949427208654714098546321163002, 8.918879626925289612669584024048, 9.778179280853780235842857791277, 10.26635763905714677762136041739