L(s) = 1 | − 2·4-s + 1.58·5-s − 3.08·7-s − 6.03·11-s − 2.36·13-s + 4·16-s − 6.85·17-s − 5.13·19-s − 3.16·20-s − 23-s − 2.50·25-s + 6.16·28-s − 29-s + 0.0784·31-s − 4.87·35-s + 5.42·37-s − 7.95·41-s − 2.05·43-s + 12.0·44-s − 1.58·47-s + 2.50·49-s + 4.73·52-s + 5.00·53-s − 9.53·55-s + 0.756·59-s − 3.79·61-s − 8·64-s + ⋯ |
L(s) = 1 | − 4-s + 0.706·5-s − 1.16·7-s − 1.81·11-s − 0.657·13-s + 16-s − 1.66·17-s − 1.17·19-s − 0.706·20-s − 0.208·23-s − 0.500·25-s + 1.16·28-s − 0.185·29-s + 0.0140·31-s − 0.823·35-s + 0.892·37-s − 1.24·41-s − 0.313·43-s + 1.81·44-s − 0.230·47-s + 0.357·49-s + 0.657·52-s + 0.687·53-s − 1.28·55-s + 0.0984·59-s − 0.485·61-s − 64-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
\(L(1)\) |
\(\approx\) |
\(0.2221907334\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2221907334\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2T^{2} \) |
| 5 | \( 1 - 1.58T + 5T^{2} \) |
| 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 + 6.03T + 11T^{2} \) |
| 13 | \( 1 + 2.36T + 13T^{2} \) |
| 17 | \( 1 + 6.85T + 17T^{2} \) |
| 19 | \( 1 + 5.13T + 19T^{2} \) |
| 31 | \( 1 - 0.0784T + 31T^{2} \) |
| 37 | \( 1 - 5.42T + 37T^{2} \) |
| 41 | \( 1 + 7.95T + 41T^{2} \) |
| 43 | \( 1 + 2.05T + 43T^{2} \) |
| 47 | \( 1 + 1.58T + 47T^{2} \) |
| 53 | \( 1 - 5.00T + 53T^{2} \) |
| 59 | \( 1 - 0.756T + 59T^{2} \) |
| 61 | \( 1 + 3.79T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 2.91T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 5.48T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 2.53T + 97T^{2} \) |
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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
−8.243149818900987459298404711194, −7.40063941119786158411405988871, −6.52327438063131279540874851582, −5.94445650812570695734063225537, −5.14444525214121841935959792255, −4.57752305382724680802023816326, −3.68161850689012867057199391599, −2.67007398277643549023008782042, −2.09100152541022055736364075259, −0.22920531326414077689182805904,
0.22920531326414077689182805904, 2.09100152541022055736364075259, 2.67007398277643549023008782042, 3.68161850689012867057199391599, 4.57752305382724680802023816326, 5.14444525214121841935959792255, 5.94445650812570695734063225537, 6.52327438063131279540874851582, 7.40063941119786158411405988871, 8.243149818900987459298404711194