L(s) = 1 | + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)9-s + (1.11 − 1.53i)13-s + (−1.11 + 0.363i)17-s + (−0.809 + 0.587i)25-s + (−0.5 + 1.53i)29-s + (−0.690 + 0.951i)37-s + (0.5 + 0.363i)41-s + (0.309 − 0.951i)45-s − 49-s + (−1.80 − 0.587i)53-s + (0.5 − 0.363i)61-s + (−1.80 − 0.587i)65-s + (1.11 + 1.53i)73-s + (0.309 + 0.951i)81-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)9-s + (1.11 − 1.53i)13-s + (−1.11 + 0.363i)17-s + (−0.809 + 0.587i)25-s + (−0.5 + 1.53i)29-s + (−0.690 + 0.951i)37-s + (0.5 + 0.363i)41-s + (0.309 − 0.951i)45-s − 49-s + (−1.80 − 0.587i)53-s + (0.5 − 0.363i)61-s + (−1.80 − 0.587i)65-s + (1.11 + 1.53i)73-s + (0.309 + 0.951i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8349811355\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8349811355\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.309 + 0.951i)T \) |
good | 3 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \) |
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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
−11.25159601574478318312356520531, −10.64615219355913085322252224651, −9.576276960024668951626368229402, −8.489396176711942148396774653560, −7.978463847911908916938723159038, −6.74549520858252794415983061086, −5.48424688781297726805139371174, −4.60856107789088923068857021480, −3.43044045478236439737860899365, −1.51592311804360774136872033011,
2.04366977008997274103467555820, 3.66106938071268329555122306825, 4.40285810040452310654867563553, 6.23331572366010476200713015836, 6.73897551594214695699563904732, 7.71483788247822969985118280640, 8.992435152637034378989124339110, 9.678877323034554381387208234493, 10.89267419724049846376994750657, 11.37380657294765696979548872316