L(s) = 1 | + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)5-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)10-s + (−0.5 − 0.363i)13-s + (−0.809 − 0.587i)16-s + (0.5 + 1.53i)17-s − 20-s + (−0.809 + 0.587i)25-s − 0.618·26-s + (0.5 − 1.53i)29-s − 32-s + (1.30 + 0.951i)34-s + (1.30 + 0.951i)37-s + ⋯ |
L(s) = 1 | + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)5-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)10-s + (−0.5 − 0.363i)13-s + (−0.809 − 0.587i)16-s + (0.5 + 1.53i)17-s − 20-s + (−0.809 + 0.587i)25-s − 0.618·26-s + (0.5 − 1.53i)29-s − 32-s + (1.30 + 0.951i)34-s + (1.30 + 0.951i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.423963161\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.423963161\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.309 + 0.951i)T \) |
good | 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−10.13110540383176579835939850246, −9.500281353636004488824856649603, −8.385713789739642149276021526797, −7.63171877349717513785500929750, −6.26717098598768207066738525179, −5.59659764838152895583636657785, −4.55234985015706056675924116951, −3.91436252679634624073741527626, −2.61240193002872695376815841275, −1.23149490184701595948393562657,
2.47470684055183168209442359936, 3.28030407126397008424340129123, 4.37427395437747443119762081843, 5.30764636196956021626146305528, 6.29984054906616827132096526159, 7.24157482103488924726725352968, 7.51037078062743494332919133313, 8.735601538043212685785742143112, 9.683651154086042175510409792962, 10.76486800999752715139386487573