L(s) = 1 | − 1.86·2-s + 1.46·4-s − 0.608·5-s + 0.188·7-s + 1.00·8-s + 1.13·10-s + 1.80·11-s − 2.52·13-s − 0.351·14-s − 4.78·16-s − 0.572·17-s − 5.70·19-s − 0.888·20-s − 3.36·22-s + 23-s − 4.62·25-s + 4.69·26-s + 0.275·28-s + 29-s + 5.66·31-s + 6.89·32-s + 1.06·34-s − 0.114·35-s − 2.76·37-s + 10.6·38-s − 0.610·40-s − 5.30·41-s + ⋯ |
L(s) = 1 | − 1.31·2-s + 0.730·4-s − 0.272·5-s + 0.0713·7-s + 0.354·8-s + 0.357·10-s + 0.545·11-s − 0.699·13-s − 0.0938·14-s − 1.19·16-s − 0.138·17-s − 1.30·19-s − 0.198·20-s − 0.717·22-s + 0.208·23-s − 0.925·25-s + 0.920·26-s + 0.0521·28-s + 0.185·29-s + 1.01·31-s + 1.21·32-s + 0.182·34-s − 0.0194·35-s − 0.453·37-s + 1.72·38-s − 0.0965·40-s − 0.828·41-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
\(L(1)\) |
\(\approx\) |
\(0.5837922945\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5837922945\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.86T + 2T^{2} \) |
| 5 | \( 1 + 0.608T + 5T^{2} \) |
| 7 | \( 1 - 0.188T + 7T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 13 | \( 1 + 2.52T + 13T^{2} \) |
| 17 | \( 1 + 0.572T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 31 | \( 1 - 5.66T + 31T^{2} \) |
| 37 | \( 1 + 2.76T + 37T^{2} \) |
| 41 | \( 1 + 5.30T + 41T^{2} \) |
| 43 | \( 1 + 2.88T + 43T^{2} \) |
| 47 | \( 1 - 7.76T + 47T^{2} \) |
| 53 | \( 1 + 9.05T + 53T^{2} \) |
| 59 | \( 1 - 1.78T + 59T^{2} \) |
| 61 | \( 1 + 4.85T + 61T^{2} \) |
| 67 | \( 1 - 6.01T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 3.12T + 73T^{2} \) |
| 79 | \( 1 - 6.66T + 79T^{2} \) |
| 83 | \( 1 - 17.9T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.148196816439777316103274128232, −7.67018322710776598185543818725, −6.73272178664947456063119616733, −6.39648380915391300174792548854, −5.11067532437381701321992848979, −4.46200035256897043806480382194, −3.63460708459349525195520460495, −2.40834659887599324413012455104, −1.67077604061314462488831009190, −0.49284258480015612216479727796,
0.49284258480015612216479727796, 1.67077604061314462488831009190, 2.40834659887599324413012455104, 3.63460708459349525195520460495, 4.46200035256897043806480382194, 5.11067532437381701321992848979, 6.39648380915391300174792548854, 6.73272178664947456063119616733, 7.67018322710776598185543818725, 8.148196816439777316103274128232