| L(s) = 1 | − 3-s + 7-s − 2·9-s − 5·11-s + 3·13-s + 6·19-s − 21-s + 23-s − 5·25-s + 5·27-s − 2·29-s + 4·31-s + 5·33-s + 8·37-s − 3·39-s − 4·41-s − 5·43-s − 47-s − 6·49-s − 6·57-s + 12·59-s − 61-s − 2·63-s − 9·67-s − 69-s + 6·71-s − 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s − 1.50·11-s + 0.832·13-s + 1.37·19-s − 0.218·21-s + 0.208·23-s − 25-s + 0.962·27-s − 0.371·29-s + 0.718·31-s + 0.870·33-s + 1.31·37-s − 0.480·39-s − 0.624·41-s − 0.762·43-s − 0.145·47-s − 6/7·49-s − 0.794·57-s + 1.56·59-s − 0.128·61-s − 0.251·63-s − 1.09·67-s − 0.120·69-s + 0.712·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.094821423154051431197019289620, −7.45969541015918164990206171508, −6.49217012448215852183462355640, −5.62258013604142023964232977689, −5.34436588610156637780275722207, −4.42614386024472550569757255818, −3.29051937274300225462314950921, −2.57467609054321276411771408587, −1.27776804211448589648623636888, 0,
1.27776804211448589648623636888, 2.57467609054321276411771408587, 3.29051937274300225462314950921, 4.42614386024472550569757255818, 5.34436588610156637780275722207, 5.62258013604142023964232977689, 6.49217012448215852183462355640, 7.45969541015918164990206171508, 8.094821423154051431197019289620
