L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.5 − 0.866i)3-s − 1.00i·4-s + (0.965 − 0.258i)5-s + (−0.965 − 0.258i)6-s + (1.22 + 1.22i)7-s + (−0.707 − 0.707i)8-s + (−0.499 + 0.866i)9-s + (0.500 − 0.866i)10-s + (−0.866 + 0.500i)12-s + (0.707 − 0.707i)13-s + 1.73·14-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (−0.366 + 0.366i)17-s + (0.258 + 0.965i)18-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.5 − 0.866i)3-s − 1.00i·4-s + (0.965 − 0.258i)5-s + (−0.965 − 0.258i)6-s + (1.22 + 1.22i)7-s + (−0.707 − 0.707i)8-s + (−0.499 + 0.866i)9-s + (0.500 − 0.866i)10-s + (−0.866 + 0.500i)12-s + (0.707 − 0.707i)13-s + 1.73·14-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (−0.366 + 0.366i)17-s + (0.258 + 0.965i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.733236801\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.733236801\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.965 + 0.258i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1.36 - 1.36i)T - iT^{2} \) |
| 47 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.93iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 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
−9.398388484470544328579477385442, −8.673377145030356773731907196197, −7.910669926970538340979690947360, −6.55979576495516884824214158708, −5.88594718310342717552165782798, −5.34447396300248130685596008556, −4.71048101636641561320595994996, −3.06040617011175151604528881591, −1.98282245917333734908314013151, −1.49971641003202065210466154496,
1.80215222937563643609871396153, 3.41808969443268059892844770801, 4.16672297679691547199713939890, 5.01848154404224133268813955116, 5.52506191665954577852216988054, 6.68759641050560193952581999319, 7.03176778719948258444633680148, 8.310471089294570348163784244725, 8.944812435876339706867441479769, 9.941340726691437993569166864159