| L(s) = 1 | + 2.80·3-s + 4.50·7-s + 4.84·9-s − 4.10·11-s − 4.10·13-s − 2.26·17-s + 6.77·19-s + 12.6·21-s + 23-s + 5.17·27-s + 4.13·29-s + 1.84·31-s − 11.4·33-s + 11.1·37-s − 11.4·39-s + 8.36·41-s + 5.43·43-s − 0.593·47-s + 13.3·49-s − 6.34·51-s + 1.70·53-s + 18.9·57-s + 6.19·59-s − 11.3·61-s + 21.8·63-s − 5.78·67-s + 2.80·69-s + ⋯ |
| L(s) = 1 | + 1.61·3-s + 1.70·7-s + 1.61·9-s − 1.23·11-s − 1.13·13-s − 0.549·17-s + 1.55·19-s + 2.75·21-s + 0.208·23-s + 0.996·27-s + 0.768·29-s + 0.330·31-s − 2.00·33-s + 1.82·37-s − 1.84·39-s + 1.30·41-s + 0.828·43-s − 0.0865·47-s + 1.90·49-s − 0.888·51-s + 0.234·53-s + 2.51·57-s + 0.806·59-s − 1.45·61-s + 2.75·63-s − 0.707·67-s + 0.337·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.696648654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.696648654\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2.80T + 3T^{2} \) |
| 7 | \( 1 - 4.50T + 7T^{2} \) |
| 11 | \( 1 + 4.10T + 11T^{2} \) |
| 13 | \( 1 + 4.10T + 13T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 19 | \( 1 - 6.77T + 19T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 - 8.36T + 41T^{2} \) |
| 43 | \( 1 - 5.43T + 43T^{2} \) |
| 47 | \( 1 + 0.593T + 47T^{2} \) |
| 53 | \( 1 - 1.70T + 53T^{2} \) |
| 59 | \( 1 - 6.19T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 5.78T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 0.363T + 73T^{2} \) |
| 79 | \( 1 - 1.75T + 79T^{2} \) |
| 83 | \( 1 + 9.72T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 - 4.38T + 97T^{2} \) |
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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
−8.903149700964531537617492800636, −8.084406620432581253783257504752, −7.67654702876374869416349768879, −7.27413917355547629879471625395, −5.66032651309995252951676495125, −4.78768698981812567773505142875, −4.25839953989357289356400372101, −2.74999535350193822561448511998, −2.53797254854737393360670295588, −1.29479657866773394012442974485,
1.29479657866773394012442974485, 2.53797254854737393360670295588, 2.74999535350193822561448511998, 4.25839953989357289356400372101, 4.78768698981812567773505142875, 5.66032651309995252951676495125, 7.27413917355547629879471625395, 7.67654702876374869416349768879, 8.084406620432581253783257504752, 8.903149700964531537617492800636
