| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 6·9-s + 6·11-s + 4·13-s + 5·16-s + 2·17-s − 12·18-s − 12·22-s + 12·23-s − 8·26-s − 6·32-s − 4·34-s + 18·36-s + 12·37-s + 10·41-s − 22·43-s + 18·44-s − 24·46-s − 11·49-s + 12·52-s + 7·64-s + 6·68-s + 4·71-s − 24·72-s − 24·74-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 2·9-s + 1.80·11-s + 1.10·13-s + 5/4·16-s + 0.485·17-s − 2.82·18-s − 2.55·22-s + 2.50·23-s − 1.56·26-s − 1.06·32-s − 0.685·34-s + 3·36-s + 1.97·37-s + 1.56·41-s − 3.35·43-s + 2.71·44-s − 3.53·46-s − 1.57·49-s + 1.66·52-s + 7/8·64-s + 0.727·68-s + 0.474·71-s − 2.82·72-s − 2.78·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.371130580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.371130580\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 37 | $C_2$ | \( 1 - 12 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
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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
−9.327132942443099488292889940657, −9.268969523930399103557526195908, −8.756589647658956089813741348675, −8.243600270361515788322679763396, −7.992769062817561822054843640713, −7.51923851862097126462611989081, −6.95891168197992567889582919432, −6.89113473791067541245818082487, −6.37346096594420201943193403575, −6.30364534805736416624401933594, −5.50480551820238957054744747931, −4.89527930528891568801512236819, −4.47789279467516989001462352110, −4.00820110255291846541532862457, −3.35209437345577244747743421522, −3.23301948941011520223928105301, −2.24628599070613559527135800698, −1.53038650823556412331430630719, −1.21145552046764018783252489119, −0.920304709990402747536254469974,
0.920304709990402747536254469974, 1.21145552046764018783252489119, 1.53038650823556412331430630719, 2.24628599070613559527135800698, 3.23301948941011520223928105301, 3.35209437345577244747743421522, 4.00820110255291846541532862457, 4.47789279467516989001462352110, 4.89527930528891568801512236819, 5.50480551820238957054744747931, 6.30364534805736416624401933594, 6.37346096594420201943193403575, 6.89113473791067541245818082487, 6.95891168197992567889582919432, 7.51923851862097126462611989081, 7.992769062817561822054843640713, 8.243600270361515788322679763396, 8.756589647658956089813741348675, 9.268969523930399103557526195908, 9.327132942443099488292889940657
