L(s) = 1 | − 3-s + 0.476·7-s + 9-s − 3.34·11-s + 13-s + 17-s − 6.77·19-s − 0.476·21-s − 4.77·23-s − 27-s + 7.59·29-s + 7.93·31-s + 3.34·33-s + 7.82·37-s − 39-s + 9.46·41-s + 1.82·43-s − 9.06·47-s − 6.77·49-s − 51-s + 9.50·53-s + 6.77·57-s + 3.16·59-s − 9.59·61-s + 0.476·63-s − 1.34·67-s + 4.77·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.180·7-s + 0.333·9-s − 1.00·11-s + 0.277·13-s + 0.242·17-s − 1.55·19-s − 0.103·21-s − 0.995·23-s − 0.192·27-s + 1.41·29-s + 1.42·31-s + 0.582·33-s + 1.28·37-s − 0.160·39-s + 1.47·41-s + 0.277·43-s − 1.32·47-s − 0.967·49-s − 0.140·51-s + 1.30·53-s + 0.897·57-s + 0.411·59-s − 1.22·61-s + 0.0600·63-s − 0.164·67-s + 0.574·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 0.476T + 7T^{2} \) |
| 11 | \( 1 + 3.34T + 11T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 + 6.77T + 19T^{2} \) |
| 23 | \( 1 + 4.77T + 23T^{2} \) |
| 29 | \( 1 - 7.59T + 29T^{2} \) |
| 31 | \( 1 - 7.93T + 31T^{2} \) |
| 37 | \( 1 - 7.82T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 - 1.82T + 43T^{2} \) |
| 47 | \( 1 + 9.06T + 47T^{2} \) |
| 53 | \( 1 - 9.50T + 53T^{2} \) |
| 59 | \( 1 - 3.16T + 59T^{2} \) |
| 61 | \( 1 + 9.59T + 61T^{2} \) |
| 67 | \( 1 + 1.34T + 67T^{2} \) |
| 71 | \( 1 + 4.86T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 2.17T + 79T^{2} \) |
| 83 | \( 1 - 9.16T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 7.04T + 97T^{2} \) |
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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
−7.64534871300336389579749172667, −6.59422264044992996861628876077, −6.17200063863426957627864504018, −5.50779935593322518707527460864, −4.49476337979363732878875839719, −4.30220810689635695705442641791, −2.96639318747627023695066468304, −2.29551530298807110433205226926, −1.14494092282001473116443733280, 0,
1.14494092282001473116443733280, 2.29551530298807110433205226926, 2.96639318747627023695066468304, 4.30220810689635695705442641791, 4.49476337979363732878875839719, 5.50779935593322518707527460864, 6.17200063863426957627864504018, 6.59422264044992996861628876077, 7.64534871300336389579749172667