L(s) = 1 | + (−0.866 + 1.5i)3-s + (−0.5 − 0.866i)5-s + (−0.866 − 0.5i)7-s + (−1 − 1.73i)9-s + 1.73·15-s + (1.5 − 0.866i)21-s + (0.866 + 1.5i)23-s + (−0.499 + 0.866i)25-s + 1.73·27-s + 29-s + 0.999i·35-s + 41-s + 1.73·43-s + (−1 + 1.73i)45-s + (0.499 + 0.866i)49-s + ⋯ |
L(s) = 1 | + (−0.866 + 1.5i)3-s + (−0.5 − 0.866i)5-s + (−0.866 − 0.5i)7-s + (−1 − 1.73i)9-s + 1.73·15-s + (1.5 − 0.866i)21-s + (0.866 + 1.5i)23-s + (−0.499 + 0.866i)25-s + 1.73·27-s + 29-s + 0.999i·35-s + 41-s + 1.73·43-s + (−1 + 1.73i)45-s + (0.499 + 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6458266478\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6458266478\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
good | 3 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - 1.73T + T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + 1.73T + T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.393030097162222583668658239065, −8.976129693503583186396641634054, −7.84453930217523303237642455877, −6.93141127189119104190415844273, −5.90702926977620149401140359372, −5.31402991040925593567164498904, −4.44835271220995208030160186489, −3.90820905313702025192651192642, −3.06825678540001141763926428415, −0.872306420878543368551202482584,
0.72813443902460122158007661839, 2.34637216350343368186167014413, 2.92055652798516423600258488818, 4.29030958929265262708550854343, 5.50270074426703396038544367209, 6.24675636203387813849334724495, 6.73441257555937397827771000009, 7.28901103937944453628923985106, 8.121444407704655220593378752923, 8.900156357031772046853475391740