| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 10-s + 4·11-s − 6·13-s − 16-s + 6·17-s + 4·19-s − 20-s − 4·22-s + 4·23-s + 25-s + 6·26-s − 29-s + 8·31-s − 5·32-s − 6·34-s + 2·37-s − 4·38-s + 3·40-s − 6·41-s + 4·43-s − 4·44-s − 4·46-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s + 1.20·11-s − 1.66·13-s − 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s + 1.17·26-s − 0.185·29-s + 1.43·31-s − 0.883·32-s − 1.02·34-s + 0.328·37-s − 0.648·38-s + 0.474·40-s − 0.937·41-s + 0.609·43-s − 0.603·44-s − 0.589·46-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.082671042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.082671042\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.27013247745378, −13.81609647054210, −13.37752015556670, −12.51643654222996, −12.33152149119124, −11.70929747132195, −11.23491262187065, −10.36367252713616, −9.958305737290688, −9.721785415548018, −9.202019136712465, −8.757817415863690, −7.999458617253165, −7.626851360927630, −7.043139642000860, −6.578801593522652, −5.707440295235328, −5.160657199050522, −4.793407069631431, −4.044982387478380, −3.383336057172041, −2.694797864741158, −1.882943352694921, −1.068129927413889, −0.6848483607207381,
0.6848483607207381, 1.068129927413889, 1.882943352694921, 2.694797864741158, 3.383336057172041, 4.044982387478380, 4.793407069631431, 5.160657199050522, 5.707440295235328, 6.578801593522652, 7.043139642000860, 7.626851360927630, 7.999458617253165, 8.757817415863690, 9.202019136712465, 9.721785415548018, 9.958305737290688, 10.36367252713616, 11.23491262187065, 11.70929747132195, 12.33152149119124, 12.51643654222996, 13.37752015556670, 13.81609647054210, 14.27013247745378
