L(s) = 1 | − 2-s + 4-s + 3.88·5-s − 1.74·7-s − 8-s − 3.88·10-s − 0.798·11-s + 0.679·13-s + 1.74·14-s + 16-s − 5.58·17-s + 6.11·19-s + 3.88·20-s + 0.798·22-s + 8.82·23-s + 10.0·25-s − 0.679·26-s − 1.74·28-s − 4.65·29-s − 8.02·31-s − 32-s + 5.58·34-s − 6.76·35-s + 4.24·37-s − 6.11·38-s − 3.88·40-s + 4.20·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.73·5-s − 0.658·7-s − 0.353·8-s − 1.22·10-s − 0.240·11-s + 0.188·13-s + 0.465·14-s + 0.250·16-s − 1.35·17-s + 1.40·19-s + 0.868·20-s + 0.170·22-s + 1.84·23-s + 2.01·25-s − 0.133·26-s − 0.329·28-s − 0.865·29-s − 1.44·31-s − 0.176·32-s + 0.958·34-s − 1.14·35-s + 0.698·37-s − 0.991·38-s − 0.613·40-s + 0.656·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.809103781\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.809103781\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 - T \) |
good | 5 | \( 1 - 3.88T + 5T^{2} \) |
| 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 + 0.798T + 11T^{2} \) |
| 13 | \( 1 - 0.679T + 13T^{2} \) |
| 17 | \( 1 + 5.58T + 17T^{2} \) |
| 19 | \( 1 - 6.11T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 + 8.02T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 - 4.20T + 41T^{2} \) |
| 43 | \( 1 + 7.19T + 43T^{2} \) |
| 47 | \( 1 - 8.05T + 47T^{2} \) |
| 53 | \( 1 - 3.31T + 53T^{2} \) |
| 59 | \( 1 - 2.34T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 2.36T + 67T^{2} \) |
| 71 | \( 1 - 8.93T + 71T^{2} \) |
| 73 | \( 1 - 2.95T + 73T^{2} \) |
| 79 | \( 1 + 3.88T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−8.884158633582494936499790727412, −7.63522529310112460810361660656, −6.89491776898740452440967543408, −6.39940840399278037757927196323, −5.52784684142979128739756302039, −5.05676249713837083659666379802, −3.56488075768617051400349408734, −2.65715100525928044693517955896, −1.95300370155054731249783758595, −0.868293077201307654192878965996,
0.868293077201307654192878965996, 1.95300370155054731249783758595, 2.65715100525928044693517955896, 3.56488075768617051400349408734, 5.05676249713837083659666379802, 5.52784684142979128739756302039, 6.39940840399278037757927196323, 6.89491776898740452440967543408, 7.63522529310112460810361660656, 8.884158633582494936499790727412