L(s) = 1 | + 1.13·2-s + 2.31·3-s − 0.707·4-s − 5-s + 2.63·6-s + 7-s − 3.07·8-s + 2.36·9-s − 1.13·10-s + 1.25·11-s − 1.63·12-s − 3.17·13-s + 1.13·14-s − 2.31·15-s − 2.08·16-s − 3.86·17-s + 2.68·18-s + 6.82·19-s + 0.707·20-s + 2.31·21-s + 1.43·22-s + 3.23·23-s − 7.12·24-s + 25-s − 3.60·26-s − 1.47·27-s − 0.707·28-s + ⋯ |
L(s) = 1 | + 0.803·2-s + 1.33·3-s − 0.353·4-s − 0.447·5-s + 1.07·6-s + 0.377·7-s − 1.08·8-s + 0.788·9-s − 0.359·10-s + 0.379·11-s − 0.473·12-s − 0.880·13-s + 0.303·14-s − 0.598·15-s − 0.520·16-s − 0.937·17-s + 0.633·18-s + 1.56·19-s + 0.158·20-s + 0.505·21-s + 0.304·22-s + 0.673·23-s − 1.45·24-s + 0.200·25-s − 0.707·26-s − 0.283·27-s − 0.133·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 - 1.13T + 2T^{2} \) |
| 3 | \( 1 - 2.31T + 3T^{2} \) |
| 11 | \( 1 - 1.25T + 11T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 + 3.86T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 + 7.94T + 29T^{2} \) |
| 31 | \( 1 + 4.08T + 31T^{2} \) |
| 37 | \( 1 - 1.56T + 37T^{2} \) |
| 41 | \( 1 + 5.36T + 41T^{2} \) |
| 43 | \( 1 + 7.80T + 43T^{2} \) |
| 47 | \( 1 - 2.57T + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 6.34T + 61T^{2} \) |
| 67 | \( 1 + 5.69T + 67T^{2} \) |
| 71 | \( 1 + 5.80T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 8.12T + 79T^{2} \) |
| 83 | \( 1 - 9.36T + 83T^{2} \) |
| 89 | \( 1 + 8.94T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−7.46717332023912648026105967371, −7.05154504344310932264859863126, −5.92837278191335122224786182394, −5.12230881982682916624305731401, −4.59893379858381527104300072847, −3.75437828559792880078995833899, −3.28717643838748888190764845095, −2.56782457656364805936219671831, −1.57148355910886147126866576242, 0,
1.57148355910886147126866576242, 2.56782457656364805936219671831, 3.28717643838748888190764845095, 3.75437828559792880078995833899, 4.59893379858381527104300072847, 5.12230881982682916624305731401, 5.92837278191335122224786182394, 7.05154504344310932264859863126, 7.46717332023912648026105967371