| L(s) = 1 | − 2·2-s + 4·4-s − 34·7-s − 8·8-s − 27·11-s − 28·13-s + 68·14-s + 16·16-s − 21·17-s + 35·19-s + 54·22-s + 78·23-s + 56·26-s − 136·28-s + 120·29-s + 182·31-s − 32·32-s + 42·34-s + 146·37-s − 70·38-s − 357·41-s − 148·43-s − 108·44-s − 156·46-s + 84·47-s + 813·49-s − 112·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.83·7-s − 0.353·8-s − 0.740·11-s − 0.597·13-s + 1.29·14-s + 1/4·16-s − 0.299·17-s + 0.422·19-s + 0.523·22-s + 0.707·23-s + 0.422·26-s − 0.917·28-s + 0.768·29-s + 1.05·31-s − 0.176·32-s + 0.211·34-s + 0.648·37-s − 0.298·38-s − 1.35·41-s − 0.524·43-s − 0.370·44-s − 0.500·46-s + 0.260·47-s + 2.37·49-s − 0.298·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7262534931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7262534931\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 34 T + p^{3} T^{2} \) |
| 11 | \( 1 + 27 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 + 21 T + p^{3} T^{2} \) |
| 19 | \( 1 - 35 T + p^{3} T^{2} \) |
| 23 | \( 1 - 78 T + p^{3} T^{2} \) |
| 29 | \( 1 - 120 T + p^{3} T^{2} \) |
| 31 | \( 1 - 182 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 + 357 T + p^{3} T^{2} \) |
| 43 | \( 1 + 148 T + p^{3} T^{2} \) |
| 47 | \( 1 - 84 T + p^{3} T^{2} \) |
| 53 | \( 1 + 702 T + p^{3} T^{2} \) |
| 59 | \( 1 - 840 T + p^{3} T^{2} \) |
| 61 | \( 1 + 238 T + p^{3} T^{2} \) |
| 67 | \( 1 - 461 T + p^{3} T^{2} \) |
| 71 | \( 1 - 708 T + p^{3} T^{2} \) |
| 73 | \( 1 + 133 T + p^{3} T^{2} \) |
| 79 | \( 1 - 650 T + p^{3} T^{2} \) |
| 83 | \( 1 - 903 T + p^{3} T^{2} \) |
| 89 | \( 1 + 735 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1106 T + p^{3} T^{2} \) |
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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
−10.28543846119491521405714128976, −9.868538131292926004651125921396, −9.034452215907530928863209725497, −7.985400545735733111318168979233, −6.92063476754623356665649885589, −6.31207439677666052582819576251, −5.02191476818346715287199190670, −3.37304916126951159140475640503, −2.52828001547276346845400078409, −0.57056152698861159667438706625,
0.57056152698861159667438706625, 2.52828001547276346845400078409, 3.37304916126951159140475640503, 5.02191476818346715287199190670, 6.31207439677666052582819576251, 6.92063476754623356665649885589, 7.985400545735733111318168979233, 9.034452215907530928863209725497, 9.868538131292926004651125921396, 10.28543846119491521405714128976
