L(s) = 1 | − 1.58·2-s + 0.508·4-s + 2.90·5-s + 0.0968·7-s + 2.36·8-s − 4.60·10-s + 4.36·11-s − 3.92·13-s − 0.153·14-s − 4.75·16-s + 1.52·17-s + 5.57·19-s + 1.47·20-s − 6.91·22-s + 23-s + 3.43·25-s + 6.21·26-s + 0.0492·28-s − 29-s + 4.44·31-s + 2.81·32-s − 2.41·34-s + 0.281·35-s − 8.67·37-s − 8.82·38-s + 6.86·40-s + 3.40·41-s + ⋯ |
L(s) = 1 | − 1.11·2-s + 0.254·4-s + 1.29·5-s + 0.0365·7-s + 0.835·8-s − 1.45·10-s + 1.31·11-s − 1.08·13-s − 0.0409·14-s − 1.18·16-s + 0.370·17-s + 1.27·19-s + 0.330·20-s − 1.47·22-s + 0.208·23-s + 0.687·25-s + 1.21·26-s + 0.00930·28-s − 0.185·29-s + 0.798·31-s + 0.497·32-s − 0.414·34-s + 0.0475·35-s − 1.42·37-s − 1.43·38-s + 1.08·40-s + 0.531·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.527214434\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.527214434\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + 1.58T + 2T^{2} \) |
| 5 | \( 1 - 2.90T + 5T^{2} \) |
| 7 | \( 1 - 0.0968T + 7T^{2} \) |
| 11 | \( 1 - 4.36T + 11T^{2} \) |
| 13 | \( 1 + 3.92T + 13T^{2} \) |
| 17 | \( 1 - 1.52T + 17T^{2} \) |
| 19 | \( 1 - 5.57T + 19T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 + 8.67T + 37T^{2} \) |
| 41 | \( 1 - 3.40T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 - 8.61T + 47T^{2} \) |
| 53 | \( 1 + 2.21T + 53T^{2} \) |
| 59 | \( 1 + 8.80T + 59T^{2} \) |
| 61 | \( 1 + 1.48T + 61T^{2} \) |
| 67 | \( 1 + 2.72T + 67T^{2} \) |
| 71 | \( 1 + 2.50T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + 2.04T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.143744026886671553419373731422, −7.45265662882752060736770263522, −6.84292241704232658906422430118, −6.07679546363878966055890536404, −5.25452321158864992992023590746, −4.61055971931970882111832730513, −3.52642755262018693973056087538, −2.41188185049560535826994933972, −1.60391026063122196265725241990, −0.835995203010026539476469134314,
0.835995203010026539476469134314, 1.60391026063122196265725241990, 2.41188185049560535826994933972, 3.52642755262018693973056087538, 4.61055971931970882111832730513, 5.25452321158864992992023590746, 6.07679546363878966055890536404, 6.84292241704232658906422430118, 7.45265662882752060736770263522, 8.143744026886671553419373731422