| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 11-s − 3.26·13-s + 15-s − 5.84·17-s − 0.678·19-s − 21-s − 3.84·23-s + 25-s − 27-s + 9.75·29-s + 3.26·31-s − 33-s − 35-s − 3.90·37-s + 3.26·39-s + 2.09·41-s − 4.58·43-s − 45-s + 3.26·47-s + 49-s + 5.84·51-s + 11.0·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.301·11-s − 0.904·13-s + 0.258·15-s − 1.41·17-s − 0.155·19-s − 0.218·21-s − 0.802·23-s + 0.200·25-s − 0.192·27-s + 1.81·29-s + 0.585·31-s − 0.174·33-s − 0.169·35-s − 0.642·37-s + 0.522·39-s + 0.327·41-s − 0.699·43-s − 0.149·45-s + 0.475·47-s + 0.142·49-s + 0.818·51-s + 1.51·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.109093457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.109093457\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 3.26T + 13T^{2} \) |
| 17 | \( 1 + 5.84T + 17T^{2} \) |
| 19 | \( 1 + 0.678T + 19T^{2} \) |
| 23 | \( 1 + 3.84T + 23T^{2} \) |
| 29 | \( 1 - 9.75T + 29T^{2} \) |
| 31 | \( 1 - 3.26T + 31T^{2} \) |
| 37 | \( 1 + 3.90T + 37T^{2} \) |
| 41 | \( 1 - 2.09T + 41T^{2} \) |
| 43 | \( 1 + 4.58T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 5.94T + 59T^{2} \) |
| 61 | \( 1 - 1.84T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 - 6.52T + 73T^{2} \) |
| 79 | \( 1 + 5.07T + 79T^{2} \) |
| 83 | \( 1 + 5.84T + 83T^{2} \) |
| 89 | \( 1 + 2.58T + 89T^{2} \) |
| 97 | \( 1 + 2.77T + 97T^{2} \) |
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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
−8.357074659066644147589476336327, −7.51621270440275864636238515557, −6.80035357845288482981499885454, −6.27939462141834489169687997182, −5.25305082633065592782655049734, −4.56718100916434563148993720591, −4.06356161007081092761210213144, −2.81106687701946227884062946060, −1.91022043938964811576198517009, −0.59178774576137195655267351768,
0.59178774576137195655267351768, 1.91022043938964811576198517009, 2.81106687701946227884062946060, 4.06356161007081092761210213144, 4.56718100916434563148993720591, 5.25305082633065592782655049734, 6.27939462141834489169687997182, 6.80035357845288482981499885454, 7.51621270440275864636238515557, 8.357074659066644147589476336327
