| L(s) = 1 | + 4.58·2-s + 13·4-s − 4.58·5-s + 22.9·8-s − 21·10-s + 22.9·11-s − 44·13-s + 0.999·16-s − 64.1·17-s + 49·19-s − 59.5·20-s + 104.·22-s − 77.9·23-s − 104·25-s − 201.·26-s − 219.·29-s − 191·31-s − 178.·32-s − 294·34-s + 29·37-s + 224.·38-s − 104.·40-s + 371.·41-s + 386·43-s + 297.·44-s − 357·46-s + 64.1·47-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.62·4-s − 0.409·5-s + 1.01·8-s − 0.664·10-s + 0.628·11-s − 0.938·13-s + 0.0156·16-s − 0.915·17-s + 0.591·19-s − 0.666·20-s + 1.01·22-s − 0.706·23-s − 0.832·25-s − 1.52·26-s − 1.40·29-s − 1.10·31-s − 0.987·32-s − 1.48·34-s + 0.128·37-s + 0.958·38-s − 0.415·40-s + 1.41·41-s + 1.36·43-s + 1.02·44-s − 1.14·46-s + 0.199·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 4.58T + 8T^{2} \) |
| 5 | \( 1 + 4.58T + 125T^{2} \) |
| 11 | \( 1 - 22.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 44T + 2.19e3T^{2} \) |
| 17 | \( 1 + 64.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 49T + 6.85e3T^{2} \) |
| 23 | \( 1 + 77.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 219.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 191T + 2.97e4T^{2} \) |
| 37 | \( 1 - 29T + 5.06e4T^{2} \) |
| 41 | \( 1 - 371.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 386T + 7.95e4T^{2} \) |
| 47 | \( 1 - 64.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 128.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 705.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 128T + 2.26e5T^{2} \) |
| 67 | \( 1 + 988T + 3.00e5T^{2} \) |
| 71 | \( 1 + 811.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 266T + 3.89e5T^{2} \) |
| 79 | \( 1 - 146T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.42e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 801.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 152T + 9.12e5T^{2} \) |
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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
−8.949774064398433092843327644809, −7.56450392723294066027661394402, −7.15594505292796046407950905658, −6.02373658190345978239694176170, −5.48340214188380998757103730138, −4.30910820214776900295097438412, −3.98265814523668909716740557219, −2.82793057919064676098275582408, −1.87206544241522940156529802891, 0,
1.87206544241522940156529802891, 2.82793057919064676098275582408, 3.98265814523668909716740557219, 4.30910820214776900295097438412, 5.48340214188380998757103730138, 6.02373658190345978239694176170, 7.15594505292796046407950905658, 7.56450392723294066027661394402, 8.949774064398433092843327644809
