L(s) = 1 | + (0.809 + 0.587i)2-s + (−2.13 − 0.693i)3-s + (0.309 + 0.951i)4-s + (−0.363 + 0.264i)5-s + (−1.31 − 1.81i)6-s + (1.26 − 0.410i)7-s + (−0.309 + 0.951i)8-s + (1.64 + 1.19i)9-s − 0.449·10-s + (3.29 − 0.362i)11-s − 2.24i·12-s + (1.70 + 1.23i)13-s + (1.26 + 0.410i)14-s + (0.959 − 0.311i)15-s + (−0.809 + 0.587i)16-s + (2.44 + 3.36i)17-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (−1.23 − 0.400i)3-s + (0.154 + 0.475i)4-s + (−0.162 + 0.118i)5-s + (−0.538 − 0.741i)6-s + (0.477 − 0.155i)7-s + (−0.109 + 0.336i)8-s + (0.549 + 0.399i)9-s − 0.142·10-s + (0.994 − 0.109i)11-s − 0.647i·12-s + (0.471 + 0.342i)13-s + (0.337 + 0.109i)14-s + (0.247 − 0.0804i)15-s + (−0.202 + 0.146i)16-s + (0.593 + 0.817i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26449 + 0.494204i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26449 + 0.494204i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (-3.29 + 0.362i)T \) |
| 19 | \( 1 + (-0.334 - 4.34i)T \) |
good | 3 | \( 1 + (2.13 + 0.693i)T + (2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (0.363 - 0.264i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (-1.26 + 0.410i)T + (5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-1.70 - 1.23i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.44 - 3.36i)T + (-5.25 + 16.1i)T^{2} \) |
| 23 | \( 1 - 4.54T + 23T^{2} \) |
| 29 | \( 1 + (0.875 + 2.69i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.26 + 8.62i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.74 + 1.21i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (2.23 - 6.88i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 10.0iT - 43T^{2} \) |
| 47 | \( 1 + (0.617 - 1.89i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.722 - 0.994i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (7.39 - 2.40i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (6.42 + 8.84i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 2.99iT - 67T^{2} \) |
| 71 | \( 1 + (-2.98 - 4.11i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.18 - 1.03i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.39 - 3.91i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (7.93 + 10.9i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 1.49iT - 89T^{2} \) |
| 97 | \( 1 + (-1.31 + 1.80i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−11.48497217668370256320478204998, −10.86404393614431898435469496579, −9.552934186113008614974874918945, −8.263292546616978851974699362777, −7.36360426091602862557159255310, −6.27876066201140787742976774807, −5.86344714524899324484275612493, −4.62715467782827546391677707155, −3.58390531431456648044559467708, −1.39981978259397288216897365561,
1.05586681172385712088672301331, 3.06691134148412189430008893440, 4.53277060621231454478487713891, 5.06423077714560385782072974301, 6.13161155371960979427425750132, 6.99587140224799410932453071209, 8.486890388229533887521202626077, 9.558519729878988726471826577432, 10.55680509250158483132619467890, 11.26510683340249740341683114738