| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s − 2·9-s + 3·11-s − 12-s − 2·13-s + 16-s − 3·17-s − 2·18-s + 7·19-s + 3·22-s − 24-s − 2·26-s + 5·27-s − 6·29-s + 4·31-s + 32-s − 3·33-s − 3·34-s − 2·36-s + 8·37-s + 7·38-s + 2·39-s + 9·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s − 2/3·9-s + 0.904·11-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.727·17-s − 0.471·18-s + 1.60·19-s + 0.639·22-s − 0.204·24-s − 0.392·26-s + 0.962·27-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.522·33-s − 0.514·34-s − 1/3·36-s + 1.31·37-s + 1.13·38-s + 0.320·39-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.255138778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.255138778\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.221701692759308985116864892196, −7.924849956818651360533058343824, −7.31618067193530795960620665696, −6.33657516384953168545460383012, −5.86978936708835719235551167417, −5.00240257185710799704212008518, −4.26777098613754301708772688504, −3.25417834090155218803572132829, −2.33477820265690219389580463783, −0.899513444547971617919916680990,
0.899513444547971617919916680990, 2.33477820265690219389580463783, 3.25417834090155218803572132829, 4.26777098613754301708772688504, 5.00240257185710799704212008518, 5.86978936708835719235551167417, 6.33657516384953168545460383012, 7.31618067193530795960620665696, 7.924849956818651360533058343824, 9.221701692759308985116864892196
