| L(s) = 1 | − 2-s + 4-s + 5-s + 3·7-s − 8-s − 10-s − 3·13-s − 3·14-s + 16-s + 4·17-s + 19-s + 20-s + 25-s + 3·26-s + 3·28-s − 3·29-s + 7·31-s − 32-s − 4·34-s + 3·35-s + 8·37-s − 38-s − 40-s + 7·41-s + 6·47-s + 2·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.13·7-s − 0.353·8-s − 0.316·10-s − 0.832·13-s − 0.801·14-s + 1/4·16-s + 0.970·17-s + 0.229·19-s + 0.223·20-s + 1/5·25-s + 0.588·26-s + 0.566·28-s − 0.557·29-s + 1.25·31-s − 0.176·32-s − 0.685·34-s + 0.507·35-s + 1.31·37-s − 0.162·38-s − 0.158·40-s + 1.09·41-s + 0.875·47-s + 2/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.155444478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.155444478\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.14456338281916, −12.72062542808098, −12.15070444268449, −11.75623358242113, −11.28185169632719, −10.90817183641088, −10.17303762362510, −9.989544343697036, −9.475553533638964, −8.905396459931083, −8.463023486189221, −7.872268225181454, −7.454223743782027, −7.281346284969951, −6.209970944433981, −6.086050105873613, −5.294888489542295, −4.914455514022942, −4.345175372280141, −3.637274292435602, −2.858979287479970, −2.354659636270487, −1.861828961385137, −1.042329336397398, −0.6804448594023008,
0.6804448594023008, 1.042329336397398, 1.861828961385137, 2.354659636270487, 2.858979287479970, 3.637274292435602, 4.345175372280141, 4.914455514022942, 5.294888489542295, 6.086050105873613, 6.209970944433981, 7.281346284969951, 7.454223743782027, 7.872268225181454, 8.463023486189221, 8.905396459931083, 9.475553533638964, 9.989544343697036, 10.17303762362510, 10.90817183641088, 11.28185169632719, 11.75623358242113, 12.15070444268449, 12.72062542808098, 13.14456338281916
