| L(s) = 1 | + (1 + i)2-s + 2i·4-s + (2 + 2i)7-s + (−2 + 2i)8-s − 3i·9-s + 2·11-s + (−4 + 4i)13-s + 4i·14-s − 4·16-s + (3 − 3i)18-s − 6i·19-s + (2 + 2i)22-s + (6 − 6i)23-s − 8·26-s + (−4 + 4i)28-s + ⋯ |
| L(s) = 1 | + (0.707 + 0.707i)2-s + i·4-s + (0.755 + 0.755i)7-s + (−0.707 + 0.707i)8-s − i·9-s + 0.603·11-s + (−1.10 + 1.10i)13-s + 1.06i·14-s − 16-s + (0.707 − 0.707i)18-s − 1.37i·19-s + (0.426 + 0.426i)22-s + (1.25 − 1.25i)23-s − 1.56·26-s + (−0.755 + 0.755i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34588 + 1.06515i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34588 + 1.06515i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 - i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3iT^{2} \) |
| 7 | \( 1 + (-2 - 2i)T + 7iT^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (4 - 4i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + (-6 + 6i)T - 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (8 + 8i)T + 37iT^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + (-2 - 2i)T + 47iT^{2} \) |
| 53 | \( 1 + (4 - 4i)T - 53iT^{2} \) |
| 59 | \( 1 - 14iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 - 14iT - 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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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
−12.51050852512989233350777280494, −12.01058425573912472670383421813, −11.12039639515583195070483304497, −9.157962624803406978300336661028, −8.837796199288290520604214729907, −7.23781402617693266289741796492, −6.54540621454981498802513765360, −5.18572397500845188810684699799, −4.25156380407754398270779325137, −2.58408328013836953007760711775,
1.61606390462301028453704667568, 3.31387394047788956235381356104, 4.70715514918353254370542366371, 5.48631819650078502909857149801, 7.12512425759943109273312593627, 8.129716278031306848507660235810, 9.723013269481225824242636840883, 10.49501904596233125654467997024, 11.28395701453297447929673370677, 12.26343214044480564145329677017
