| L(s) = 1 | + 4·2-s + 2·3-s + 12·4-s + 8·6-s + 32·8-s − 33·9-s − 26·11-s + 24·12-s − 102·13-s + 80·16-s + 186·17-s − 132·18-s − 36·19-s − 104·22-s + 44·23-s + 64·24-s − 408·26-s − 86·27-s − 46·29-s − 140·31-s + 192·32-s − 52·33-s + 744·34-s − 396·36-s − 132·37-s − 144·38-s − 204·39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.384·3-s + 3/2·4-s + 0.544·6-s + 1.41·8-s − 1.22·9-s − 0.712·11-s + 0.577·12-s − 2.17·13-s + 5/4·16-s + 2.65·17-s − 1.72·18-s − 0.434·19-s − 1.00·22-s + 0.398·23-s + 0.544·24-s − 3.07·26-s − 0.612·27-s − 0.294·29-s − 0.811·31-s + 1.06·32-s − 0.274·33-s + 3.75·34-s − 1.83·36-s − 0.586·37-s − 0.614·38-s − 0.837·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 37 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 26 T + 2703 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 102 T + 6833 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 186 T + 17897 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 36 T + 13530 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 44 T + 192 p T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 46 T + 38939 T^{2} + 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 140 T + 38944 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 132 T + 28044 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 132 T + 100148 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 324 T + 181386 T^{2} + 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 242 T + 215325 T^{2} - 242 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 208 T + 205512 T^{2} + 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 780 T + 529576 T^{2} + 780 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 216 T + 425298 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 256 T + 62452 T^{2} - 256 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 692 T + 813066 T^{2} + 692 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 832 T + 948202 T^{2} - 832 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1962 T + 1889271 T^{2} + 1962 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1712 T + 1530198 T^{2} - 1712 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1960 T + 2217986 T^{2} + 1960 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 102 T + 453465 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.163641507000148339141875183756, −7.81565046565702530990522804132, −7.50750010462713995125796792231, −7.49207950948322931432668682912, −6.72733922649814139654981742406, −6.46961405235075972280301794990, −5.69470977931665623769281672315, −5.62532921831608581727023199358, −5.19078623416409261219172472354, −5.06539679406290288752027382905, −4.44008725062032927004163514328, −3.94734928015232213555292404472, −3.35260229044110577072366544651, −3.10580570142473909234121942666, −2.66052133220845455595629460905, −2.45243666330662789579185623490, −1.68979993845544675916982415993, −1.18864224243975008076719937293, 0, 0,
1.18864224243975008076719937293, 1.68979993845544675916982415993, 2.45243666330662789579185623490, 2.66052133220845455595629460905, 3.10580570142473909234121942666, 3.35260229044110577072366544651, 3.94734928015232213555292404472, 4.44008725062032927004163514328, 5.06539679406290288752027382905, 5.19078623416409261219172472354, 5.62532921831608581727023199358, 5.69470977931665623769281672315, 6.46961405235075972280301794990, 6.72733922649814139654981742406, 7.49207950948322931432668682912, 7.50750010462713995125796792231, 7.81565046565702530990522804132, 8.163641507000148339141875183756
