L(s) = 1 | + (−0.221 + 0.221i)2-s + 0.902i·4-s + (−0.809 + 0.587i)5-s + (−1.26 + 1.26i)7-s + (−0.420 − 0.420i)8-s + i·9-s + (0.0489 − 0.309i)10-s + 1.90·11-s − 0.557i·14-s − 0.715·16-s + (−0.221 − 0.221i)18-s + (−0.530 − 0.729i)20-s + (−0.420 + 0.420i)22-s + (0.309 − 0.951i)25-s + (−1.13 − 1.13i)28-s + 1.61i·29-s + ⋯ |
L(s) = 1 | + (−0.221 + 0.221i)2-s + 0.902i·4-s + (−0.809 + 0.587i)5-s + (−1.26 + 1.26i)7-s + (−0.420 − 0.420i)8-s + i·9-s + (0.0489 − 0.309i)10-s + 1.90·11-s − 0.557i·14-s − 0.715·16-s + (−0.221 − 0.221i)18-s + (−0.530 − 0.729i)20-s + (−0.420 + 0.420i)22-s + (0.309 − 0.951i)25-s + (−1.13 − 1.13i)28-s + 1.61i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6436442907\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6436442907\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.809 - 0.587i)T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 + (0.221 - 0.221i)T - iT^{2} \) |
| 3 | \( 1 - iT^{2} \) |
| 7 | \( 1 + (1.26 - 1.26i)T - iT^{2} \) |
| 11 | \( 1 - 1.90T + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - 1.61iT - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + 1.17T + T^{2} \) |
| 43 | \( 1 + (1.26 + 1.26i)T + iT^{2} \) |
| 47 | \( 1 + (-0.221 + 0.221i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.221 - 0.221i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (-1 - i)T + iT^{2} \) |
| 89 | \( 1 + 1.90iT - T^{2} \) |
| 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
−9.501812733108901561795262767516, −8.736035482258990262967791009762, −8.435248736664700291600643860658, −7.19289616368012617379592877014, −6.83166704674457527355506779711, −6.08104417921846240972233850667, −4.84553566646704005869962753647, −3.62026032227337253544220732873, −3.29839925249418440758827739939, −2.15253366149485497566265419712,
0.53657553587055374050787212647, 1.38964612056731521837406879468, 3.30393388211978263788813316756, 3.93110282302529875702259876435, 4.62518422797022208444440264964, 6.09404552042216246274973685327, 6.52683271736686964326739643977, 7.17440259158546449949241436748, 8.364663925314052524823332067168, 9.270202741791219067418993373840