L(s) = 1 | + (0.221 − 0.221i)3-s + (1.26 + 1.26i)7-s + 0.902i·9-s + 0.557·21-s + (0.642 − 0.642i)23-s + (0.420 + 0.420i)27-s − 1.61i·29-s − 1.17·41-s + (−1.39 + 1.39i)43-s + (1.39 + 1.39i)47-s + 2.17i·49-s − 1.90·61-s + (−1.13 + 1.13i)63-s + (1 + i)67-s − 0.284i·69-s + ⋯ |
L(s) = 1 | + (0.221 − 0.221i)3-s + (1.26 + 1.26i)7-s + 0.902i·9-s + 0.557·21-s + (0.642 − 0.642i)23-s + (0.420 + 0.420i)27-s − 1.61i·29-s − 1.17·41-s + (−1.39 + 1.39i)43-s + (1.39 + 1.39i)47-s + 2.17i·49-s − 1.90·61-s + (−1.13 + 1.13i)63-s + (1 + i)67-s − 0.284i·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.624462621\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.624462621\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.221 + 0.221i)T - iT^{2} \) |
| 7 | \( 1 + (-1.26 - 1.26i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.642 + 0.642i)T - iT^{2} \) |
| 29 | \( 1 + 1.61iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + 1.17T + T^{2} \) |
| 43 | \( 1 + (1.39 - 1.39i)T - iT^{2} \) |
| 47 | \( 1 + (-1.39 - 1.39i)T + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.90T + T^{2} \) |
| 67 | \( 1 + (-1 - i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-1.26 + 1.26i)T - iT^{2} \) |
| 89 | \( 1 + 0.618iT - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−8.610494648800721604879542690694, −7.996956592027485703026404130788, −7.55973518389088543401967996022, −6.43010709491117415246423493199, −5.70495714292999236466799728563, −4.90046856882448747359728260148, −4.46524265923008815291780177896, −2.98799035357342695566418471572, −2.28956898046998049278417151306, −1.51122636176384452130260709305,
0.975156422984974655631985791947, 1.90343164351988354093423999321, 3.39727445270167248855976893761, 3.78662983558404878551391607778, 4.83464549328545117264910305144, 5.28484151066663431212702762802, 6.56219048812739275590677879059, 7.11362745063964175260237132487, 7.75614858012828256769888070842, 8.633808838135999941808845730093