| L(s) = 1 | − 3-s + 2·7-s + 9-s − 4·11-s + 13-s − 8·17-s + 6·19-s − 2·21-s + 6·23-s − 27-s − 4·29-s + 4·33-s + 2·37-s − 39-s − 2·41-s − 4·43-s − 3·49-s + 8·51-s + 10·53-s − 6·57-s − 4·59-s − 10·61-s + 2·63-s + 12·67-s − 6·69-s + 8·71-s + 8·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 1.94·17-s + 1.37·19-s − 0.436·21-s + 1.25·23-s − 0.192·27-s − 0.742·29-s + 0.696·33-s + 0.328·37-s − 0.160·39-s − 0.312·41-s − 0.609·43-s − 3/7·49-s + 1.12·51-s + 1.37·53-s − 0.794·57-s − 0.520·59-s − 1.28·61-s + 0.251·63-s + 1.46·67-s − 0.722·69-s + 0.949·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−16.16512718201445, −15.75008352975929, −15.20690290447186, −14.85360899637116, −13.80651280796235, −13.53128307575016, −12.98169348406460, −12.40815612736098, −11.58323463549276, −11.11500501193250, −10.93279981149979, −10.15184100743799, −9.443267080078740, −8.857365854034218, −8.181842289383246, −7.578452561779419, −6.986171915811535, −6.386193497760841, −5.452152088942617, −5.095956048181121, −4.555912580334454, −3.659154986738195, −2.753566817548273, −2.014965545083047, −1.070478590551753, 0,
1.070478590551753, 2.014965545083047, 2.753566817548273, 3.659154986738195, 4.555912580334454, 5.095956048181121, 5.452152088942617, 6.386193497760841, 6.986171915811535, 7.578452561779419, 8.181842289383246, 8.857365854034218, 9.443267080078740, 10.15184100743799, 10.93279981149979, 11.11500501193250, 11.58323463549276, 12.40815612736098, 12.98169348406460, 13.53128307575016, 13.80651280796235, 14.85360899637116, 15.20690290447186, 15.75008352975929, 16.16512718201445
