L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s + 16-s − 4·17-s − 18-s − 19-s − 4·23-s − 24-s − 5·25-s + 27-s + 2·29-s + 8·31-s − 32-s + 4·34-s + 36-s + 10·37-s + 38-s − 8·41-s − 4·43-s + 4·46-s + 8·47-s + 48-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.229·19-s − 0.834·23-s − 0.204·24-s − 25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.176·32-s + 0.685·34-s + 1/6·36-s + 1.64·37-s + 0.162·38-s − 1.24·41-s − 0.609·43-s + 0.589·46-s + 1.16·47-s + 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.928367267161403711977385220538, −7.27952836900128359179075955551, −6.41910081172113472811412925931, −5.93882590944141446563503152406, −4.68556461145771088251370923737, −4.08337686555717169288180656080, −3.00940921087126050439565110472, −2.29813396887053661837209937324, −1.39360492693941603418836725105, 0,
1.39360492693941603418836725105, 2.29813396887053661837209937324, 3.00940921087126050439565110472, 4.08337686555717169288180656080, 4.68556461145771088251370923737, 5.93882590944141446563503152406, 6.41910081172113472811412925931, 7.27952836900128359179075955551, 7.928367267161403711977385220538