L(s) = 1 | + 3-s − 1.65·7-s + 9-s − 2.94·11-s − 13-s + 1.46·17-s − 1.65·21-s + 0.532·23-s + 27-s + 5.70·29-s − 2.94·33-s + 8.77·37-s − 39-s + 1.23·41-s − 1.70·43-s + 2.70·47-s − 4.25·49-s + 1.46·51-s − 8.77·53-s − 3.83·59-s − 0.241·61-s − 1.65·63-s + 2.58·67-s + 0.532·69-s − 2.55·71-s + 0.188·73-s + 4.88·77-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.625·7-s + 0.333·9-s − 0.888·11-s − 0.277·13-s + 0.355·17-s − 0.361·21-s + 0.111·23-s + 0.192·27-s + 1.06·29-s − 0.513·33-s + 1.44·37-s − 0.160·39-s + 0.193·41-s − 0.260·43-s + 0.394·47-s − 0.608·49-s + 0.205·51-s − 1.20·53-s − 0.498·59-s − 0.0308·61-s − 0.208·63-s + 0.315·67-s + 0.0641·69-s − 0.302·71-s + 0.0220·73-s + 0.556·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.015111531\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.015111531\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.65T + 7T^{2} \) |
| 11 | \( 1 + 2.94T + 11T^{2} \) |
| 17 | \( 1 - 1.46T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 0.532T + 23T^{2} \) |
| 29 | \( 1 - 5.70T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 8.77T + 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 + 1.70T + 43T^{2} \) |
| 47 | \( 1 - 2.70T + 47T^{2} \) |
| 53 | \( 1 + 8.77T + 53T^{2} \) |
| 59 | \( 1 + 3.83T + 59T^{2} \) |
| 61 | \( 1 + 0.241T + 61T^{2} \) |
| 67 | \( 1 - 2.58T + 67T^{2} \) |
| 71 | \( 1 + 2.55T + 71T^{2} \) |
| 73 | \( 1 - 0.188T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 7.91T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−7.82881347557312725847228154062, −7.33121519789223046603713778690, −6.46050738205326956522775181044, −5.87647149054203094835792208456, −4.92609536109393631306848402345, −4.34570990013611596337739047722, −3.25404561965378132209991954336, −2.85889082540437103522130234414, −1.92930650299947639460633571241, −0.67012578258340910239716362452,
0.67012578258340910239716362452, 1.92930650299947639460633571241, 2.85889082540437103522130234414, 3.25404561965378132209991954336, 4.34570990013611596337739047722, 4.92609536109393631306848402345, 5.87647149054203094835792208456, 6.46050738205326956522775181044, 7.33121519789223046603713778690, 7.82881347557312725847228154062