L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s + 2·11-s + 4·13-s − 2·14-s + 16-s + 6·17-s + 19-s − 2·22-s + 8·23-s − 4·26-s + 2·28-s + 6·29-s − 8·31-s − 32-s − 6·34-s + 8·37-s − 38-s + 12·41-s + 2·44-s − 8·46-s − 3·49-s + 4·52-s + 10·53-s − 2·56-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.603·11-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.229·19-s − 0.426·22-s + 1.66·23-s − 0.784·26-s + 0.377·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 1.31·37-s − 0.162·38-s + 1.87·41-s + 0.301·44-s − 1.17·46-s − 3/7·49-s + 0.554·52-s + 1.37·53-s − 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.283903445\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.283903445\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.63225226432738688177430953039, −7.44429826682966942118891439317, −6.42271945035508711397145967505, −5.83385252613455516880386302754, −5.10731987798687737988201795288, −4.18872519421385749150798113526, −3.38315328598347331482176585872, −2.58796586156529729200888485667, −1.31941914547321963718406538015, −1.01291769438556015766313322328,
1.01291769438556015766313322328, 1.31941914547321963718406538015, 2.58796586156529729200888485667, 3.38315328598347331482176585872, 4.18872519421385749150798113526, 5.10731987798687737988201795288, 5.83385252613455516880386302754, 6.42271945035508711397145967505, 7.44429826682966942118891439317, 7.63225226432738688177430953039