L(s) = 1 | + 2·3-s − 2·5-s + 9-s + 4·11-s − 6·13-s − 4·15-s − 4·17-s + 6·19-s + 4·23-s − 25-s − 4·27-s + 6·29-s + 4·31-s + 8·33-s + 6·37-s − 12·39-s + 4·41-s + 12·43-s − 2·45-s + 12·47-s − 8·51-s − 6·53-s − 8·55-s + 12·57-s + 6·59-s + 6·61-s + 12·65-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 1.03·15-s − 0.970·17-s + 1.37·19-s + 0.834·23-s − 1/5·25-s − 0.769·27-s + 1.11·29-s + 0.718·31-s + 1.39·33-s + 0.986·37-s − 1.92·39-s + 0.624·41-s + 1.82·43-s − 0.298·45-s + 1.75·47-s − 1.12·51-s − 0.824·53-s − 1.07·55-s + 1.58·57-s + 0.781·59-s + 0.768·61-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.315738732\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.315738732\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.683081961552536023874134503712, −7.925598430243252779527582218255, −7.34799266760693105721346835931, −6.77035761879384889307893847964, −5.57878625800576661743532675218, −4.46885810205421187130516552810, −4.01305938724372849329590700559, −2.93470599629258036794766312355, −2.40706782693886066037265071101, −0.873856170039672848452509089971,
0.873856170039672848452509089971, 2.40706782693886066037265071101, 2.93470599629258036794766312355, 4.01305938724372849329590700559, 4.46885810205421187130516552810, 5.57878625800576661743532675218, 6.77035761879384889307893847964, 7.34799266760693105721346835931, 7.925598430243252779527582218255, 8.683081961552536023874134503712