| L(s) = 1 | + 4-s − 2·5-s − 2·9-s − 4·11-s − 3·16-s + 12·19-s − 2·20-s − 25-s + 4·29-s − 8·31-s − 2·36-s − 4·44-s + 4·45-s − 2·49-s + 8·55-s + 24·61-s − 7·64-s − 4·71-s + 12·76-s + 7·79-s + 6·80-s − 5·81-s + 4·89-s − 24·95-s + 8·99-s − 100-s − 4·101-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s − 2/3·9-s − 1.20·11-s − 3/4·16-s + 2.75·19-s − 0.447·20-s − 1/5·25-s + 0.742·29-s − 1.43·31-s − 1/3·36-s − 0.603·44-s + 0.596·45-s − 2/7·49-s + 1.07·55-s + 3.07·61-s − 7/8·64-s − 0.474·71-s + 1.37·76-s + 0.787·79-s + 0.670·80-s − 5/9·81-s + 0.423·89-s − 2.46·95-s + 0.804·99-s − 0.0999·100-s − 0.398·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1975 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1975 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6025581749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6025581749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 8 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.37986530846954705950056593225, −12.67291314829937564288649221437, −11.87599027468249652033453316468, −11.51753330119097474668348955272, −11.11823759886520028697085792314, −10.26445603410347596789046697551, −9.578997724649700768574424691141, −8.784882939553097875023304040321, −7.929855807415493542595970934579, −7.51433060362708187443102304743, −6.82019914774084213414421945202, −5.59522213311270144437373420326, −5.06971336265719005763494136696, −3.67174885318377485777768194381, −2.70351313949228236279027148118,
2.70351313949228236279027148118, 3.67174885318377485777768194381, 5.06971336265719005763494136696, 5.59522213311270144437373420326, 6.82019914774084213414421945202, 7.51433060362708187443102304743, 7.929855807415493542595970934579, 8.784882939553097875023304040321, 9.578997724649700768574424691141, 10.26445603410347596789046697551, 11.11823759886520028697085792314, 11.51753330119097474668348955272, 11.87599027468249652033453316468, 12.67291314829937564288649221437, 13.37986530846954705950056593225
