L(s) = 1 | + 3-s − 6·5-s + 2·7-s − 2·9-s − 6·15-s + 12·17-s + 2·21-s + 17·25-s − 5·27-s − 12·35-s − 2·37-s − 20·43-s + 12·45-s − 3·49-s + 12·51-s + 6·59-s − 4·63-s − 2·67-s + 17·75-s + 4·79-s + 81-s + 12·83-s − 72·85-s − 18·89-s + 36·101-s − 12·105-s + 4·109-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 2.68·5-s + 0.755·7-s − 2/3·9-s − 1.54·15-s + 2.91·17-s + 0.436·21-s + 17/5·25-s − 0.962·27-s − 2.02·35-s − 0.328·37-s − 3.04·43-s + 1.78·45-s − 3/7·49-s + 1.68·51-s + 0.781·59-s − 0.503·63-s − 0.244·67-s + 1.96·75-s + 0.450·79-s + 1/9·81-s + 1.31·83-s − 7.80·85-s − 1.90·89-s + 3.58·101-s − 1.17·105-s + 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 853776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 853776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.030135611649600137523792877956, −7.60852684015633966980596452192, −7.57107715370897119251299700280, −6.85141867351762882928486006631, −6.29465577828163423751170725006, −5.55064102283955097263086227457, −5.11227132195889762323685903220, −4.80957445265205313059818308287, −3.98043613480578289233430491973, −3.69194524730832991586322636375, −3.27381985666804460223319703873, −3.00610279743917756108045023038, −1.87692294389813555618766341906, −1.00356671220607536314623203552, 0,
1.00356671220607536314623203552, 1.87692294389813555618766341906, 3.00610279743917756108045023038, 3.27381985666804460223319703873, 3.69194524730832991586322636375, 3.98043613480578289233430491973, 4.80957445265205313059818308287, 5.11227132195889762323685903220, 5.55064102283955097263086227457, 6.29465577828163423751170725006, 6.85141867351762882928486006631, 7.57107715370897119251299700280, 7.60852684015633966980596452192, 8.030135611649600137523792877956