L(s) = 1 | + 2.60i·2-s − i·3-s − 4.77·4-s + i·5-s + 2.60·6-s + (−2.60 + 0.474i)7-s − 7.22i·8-s − 9-s − 2.60·10-s + (3.23 + 0.726i)11-s + 4.77i·12-s − 6.07·13-s + (−1.23 − 6.77i)14-s + 15-s + 9.24·16-s − 4.27·17-s + ⋯ |
L(s) = 1 | + 1.84i·2-s − 0.577i·3-s − 2.38·4-s + 0.447i·5-s + 1.06·6-s + (−0.983 + 0.179i)7-s − 2.55i·8-s − 0.333·9-s − 0.823·10-s + (0.975 + 0.219i)11-s + 1.37i·12-s − 1.68·13-s + (−0.330 − 1.81i)14-s + 0.258·15-s + 2.31·16-s − 1.03·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4942073038\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4942073038\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (2.60 - 0.474i)T \) |
| 11 | \( 1 + (-3.23 - 0.726i)T \) |
good | 2 | \( 1 - 2.60iT - 2T^{2} \) |
| 13 | \( 1 + 6.07T + 13T^{2} \) |
| 17 | \( 1 + 4.27T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 5.77T + 23T^{2} \) |
| 29 | \( 1 + 6.49iT - 29T^{2} \) |
| 31 | \( 1 + 2.66iT - 31T^{2} \) |
| 37 | \( 1 - 2.66T + 37T^{2} \) |
| 41 | \( 1 - 8.37T + 41T^{2} \) |
| 43 | \( 1 + 11.9iT - 43T^{2} \) |
| 47 | \( 1 + 12.6iT - 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 4.89iT - 59T^{2} \) |
| 61 | \( 1 + 7.86T + 61T^{2} \) |
| 67 | \( 1 - 2.92T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 4.53T + 73T^{2} \) |
| 79 | \( 1 - 7.42iT - 79T^{2} \) |
| 83 | \( 1 - 9.31T + 83T^{2} \) |
| 89 | \( 1 - 0.418iT - 89T^{2} \) |
| 97 | \( 1 - 2.43iT - 97T^{2} \) |
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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
−9.330808031729991207522070158505, −8.963042247637569977454057253488, −7.69989926367120764442601687336, −7.15526358117494338135463251081, −6.63895593342753211527414052910, −5.93240478391794063457577966030, −4.95476827427470876677107799492, −3.94480202929249222087679059037, −2.55313041173856851520477604993, −0.24066639516278174519600788012,
1.20537364861600472949678927314, 2.72155408546757369413322081673, 3.30374221365016117182495848126, 4.49643501465602060857831025949, 4.85055848726916521341904335487, 6.26781692403086196654124543707, 7.50491829891143593937627979503, 8.893057086804726056933313523946, 9.415853698412273047624178637660, 9.636366518798445902020686756235