L(s) = 1 | + (1 + i)2-s + i·4-s + 5-s + (−1 − i)7-s − i·9-s + (1 + i)10-s − 2i·14-s + 16-s + (1 − i)18-s + i·20-s + 25-s + (1 − i)28-s + 2i·29-s + (1 + i)32-s + (−1 − i)35-s + 36-s + ⋯ |
L(s) = 1 | + (1 + i)2-s + i·4-s + 5-s + (−1 − i)7-s − i·9-s + (1 + i)10-s − 2i·14-s + 16-s + (1 − i)18-s + i·20-s + 25-s + (1 − i)28-s + 2i·29-s + (1 + i)32-s + (−1 − i)35-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.143417792\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.143417792\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 + (-1 - i)T + iT^{2} \) |
| 3 | \( 1 + iT^{2} \) |
| 7 | \( 1 + (1 + i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - 2iT - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 + (1 + i)T + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1 - i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (-1 + i)T - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−9.464182653012213707342728886500, −8.589029666025352436802912103825, −7.37957451693754690946929877592, −6.67178073060197634594904605399, −6.45189275431516334361112919265, −5.52130334437881394348948072526, −4.73754542577102845056180279032, −3.68191271787778485252461221164, −3.12026842389227566809209836210, −1.29255610354854101434201687764,
1.81798979340279615411955186206, 2.49167989300290052548453385176, 3.16664143017195651785313608554, 4.36086444443641520119793377363, 5.22585919524142474087844407206, 5.82117491680344010526795957342, 6.51510200986283288967670550136, 7.78353551958036844312843160141, 8.667637977783349476637630414117, 9.643087538873229781352202371049