L(s) = 1 | + (0.974 + 0.222i)5-s + (−0.222 − 0.974i)9-s + (−0.119 − 0.189i)13-s − 1.80i·17-s + (0.900 + 0.433i)25-s + (0.974 + 0.222i)29-s + (−0.347 − 1.52i)37-s + (0.158 + 0.158i)41-s − i·45-s + (−0.974 + 0.222i)49-s + (0.623 + 1.78i)53-s + (0.211 + 1.87i)61-s + (−0.0739 − 0.211i)65-s + (0.376 + 0.781i)73-s + (−0.900 + 0.433i)81-s + ⋯ |
L(s) = 1 | + (0.974 + 0.222i)5-s + (−0.222 − 0.974i)9-s + (−0.119 − 0.189i)13-s − 1.80i·17-s + (0.900 + 0.433i)25-s + (0.974 + 0.222i)29-s + (−0.347 − 1.52i)37-s + (0.158 + 0.158i)41-s − i·45-s + (−0.974 + 0.222i)49-s + (0.623 + 1.78i)53-s + (0.211 + 1.87i)61-s + (−0.0739 − 0.211i)65-s + (0.376 + 0.781i)73-s + (−0.900 + 0.433i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.381773734\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.381773734\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.974 - 0.222i)T \) |
| 29 | \( 1 + (-0.974 - 0.222i)T \) |
good | 3 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 7 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 11 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 13 | \( 1 + (0.119 + 0.189i)T + (-0.433 + 0.900i)T^{2} \) |
| 17 | \( 1 + 1.80iT - T^{2} \) |
| 19 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 23 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 31 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 37 | \( 1 + (0.347 + 1.52i)T + (-0.900 + 0.433i)T^{2} \) |
| 41 | \( 1 + (-0.158 - 0.158i)T + iT^{2} \) |
| 43 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 53 | \( 1 + (-0.623 - 1.78i)T + (-0.781 + 0.623i)T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + (-0.211 - 1.87i)T + (-0.974 + 0.222i)T^{2} \) |
| 67 | \( 1 + (0.433 + 0.900i)T^{2} \) |
| 71 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.376 - 0.781i)T + (-0.623 + 0.781i)T^{2} \) |
| 79 | \( 1 + (0.433 + 0.900i)T^{2} \) |
| 83 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 89 | \( 1 + (-0.559 - 1.59i)T + (-0.781 + 0.623i)T^{2} \) |
| 97 | \( 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)T^{2} \) |
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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
−9.256767010784778255577000805377, −8.557645414180073031718809401093, −7.34277637689946458873042490574, −6.83617261331328621469753493756, −5.94353075089158949512531555343, −5.32353207847349761135160418543, −4.33900021730659504518331537372, −3.11554565705152443355776235894, −2.48853997705899974942694883237, −1.02911587843854396424917622425,
1.56406536340793072306094015850, 2.31060697843739983234746931459, 3.48220751748494800270161045866, 4.67404872172843023128626346966, 5.25759919102729838813644080548, 6.21062590916149497090557714506, 6.69995591781026019336729893348, 8.027865115272372169526393091402, 8.348003785044959787802868043901, 9.254897606206087200459630134379