L(s) = 1 | − 2.17·2-s + 2.70·4-s + 3.70·7-s − 1.53·8-s − 11-s − 1.70·13-s − 8.04·14-s − 2.07·16-s + 6.04·17-s − 3.07·19-s + 2.17·22-s − 4·23-s + 3.70·26-s + 10.0·28-s − 5.26·29-s − 6.34·31-s + 7.58·32-s − 13.1·34-s − 3.41·37-s + 6.68·38-s − 9.57·41-s − 3.12·43-s − 2.70·44-s + 8.68·46-s − 2.73·47-s + 6.75·49-s − 4.63·52-s + ⋯ |
L(s) = 1 | − 1.53·2-s + 1.35·4-s + 1.40·7-s − 0.544·8-s − 0.301·11-s − 0.474·13-s − 2.15·14-s − 0.519·16-s + 1.46·17-s − 0.706·19-s + 0.462·22-s − 0.834·23-s + 0.727·26-s + 1.89·28-s − 0.977·29-s − 1.13·31-s + 1.34·32-s − 2.25·34-s − 0.562·37-s + 1.08·38-s − 1.49·41-s − 0.476·43-s − 0.408·44-s + 1.27·46-s − 0.399·47-s + 0.965·49-s − 0.642·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 7 | \( 1 - 3.70T + 7T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 17 | \( 1 - 6.04T + 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 5.26T + 29T^{2} \) |
| 31 | \( 1 + 6.34T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 + 9.57T + 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 + 2.73T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 3.60T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 1.84T + 67T^{2} \) |
| 71 | \( 1 - 7.23T + 71T^{2} \) |
| 73 | \( 1 - 6.38T + 73T^{2} \) |
| 79 | \( 1 + 7.44T + 79T^{2} \) |
| 83 | \( 1 - 7.86T + 83T^{2} \) |
| 89 | \( 1 - 5.02T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−8.468011384803371787375726442353, −7.81920611287335545793371627043, −7.57709793800404457020133761072, −6.51545925301593210653588182068, −5.40745465744103171265371141592, −4.72399889439661067093111268191, −3.48270238010063447432081845998, −2.04784110836699239653169217130, −1.50847258840660794552701925734, 0,
1.50847258840660794552701925734, 2.04784110836699239653169217130, 3.48270238010063447432081845998, 4.72399889439661067093111268191, 5.40745465744103171265371141592, 6.51545925301593210653588182068, 7.57709793800404457020133761072, 7.81920611287335545793371627043, 8.468011384803371787375726442353