| L(s) = 1 | + 4·2-s + 12·4-s + 10·5-s + 7·7-s + 32·8-s + 40·10-s − 15·11-s + 49·13-s + 28·14-s + 80·16-s + 24·17-s + 64·19-s + 120·20-s − 60·22-s + 147·23-s + 75·25-s + 196·26-s + 84·28-s + 69·29-s − 143·31-s + 192·32-s + 96·34-s + 70·35-s + 358·37-s + 256·38-s + 320·40-s + 432·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 0.377·7-s + 1.41·8-s + 1.26·10-s − 0.411·11-s + 1.04·13-s + 0.534·14-s + 5/4·16-s + 0.342·17-s + 0.772·19-s + 1.34·20-s − 0.581·22-s + 1.33·23-s + 3/5·25-s + 1.47·26-s + 0.566·28-s + 0.441·29-s − 0.828·31-s + 1.06·32-s + 0.484·34-s + 0.338·35-s + 1.59·37-s + 1.09·38-s + 1.26·40-s + 1.64·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(15.16400383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.16400383\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - p T + 576 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 15 T - 338 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 49 T + 1938 T^{2} - 49 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 24 T + 8014 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 64 T + 14253 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 147 T + 14944 T^{2} - 147 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 69 T - 14702 T^{2} - 69 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 143 T + 61638 T^{2} + 143 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 358 T + 128946 T^{2} - 358 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 432 T + 144889 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 628 T + 208710 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 225 T + 117490 T^{2} + 225 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 231 T + 17572 T^{2} + 231 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 744 T + 548653 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 802 T + 575154 T^{2} - 802 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 206 T + 502110 T^{2} + 206 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1329 T + 971440 T^{2} + 1329 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 244 T + 722502 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1474 T + 1172766 T^{2} - 1474 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 858 T + 1268446 T^{2} - 858 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2517 T + 2992660 T^{2} + 2517 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1852 T + 2352258 T^{2} - 1852 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.07438874302542202008744128998, −9.774897844011206267141192345136, −9.207205322514338363401306904742, −8.946475344405313666666188683243, −8.151786927578990768251769605036, −7.930804116309785510129656659007, −7.24790624473775505252830181315, −7.06945638269254241064278395559, −6.20625947588879049741614779515, −6.16311492247961819957840338806, −5.44440010091339931195086914071, −5.41964634173351937024470366118, −4.53998309516508552261515397379, −4.46121285021217709986845157901, −3.51593436912574619994812887486, −3.25895097925843646336310079076, −2.50031870462329804004331364341, −2.19409978432153863832560539908, −1.16433990892102177576338590541, −0.961860842499966177569457055613,
0.961860842499966177569457055613, 1.16433990892102177576338590541, 2.19409978432153863832560539908, 2.50031870462329804004331364341, 3.25895097925843646336310079076, 3.51593436912574619994812887486, 4.46121285021217709986845157901, 4.53998309516508552261515397379, 5.41964634173351937024470366118, 5.44440010091339931195086914071, 6.16311492247961819957840338806, 6.20625947588879049741614779515, 7.06945638269254241064278395559, 7.24790624473775505252830181315, 7.930804116309785510129656659007, 8.151786927578990768251769605036, 8.946475344405313666666188683243, 9.207205322514338363401306904742, 9.774897844011206267141192345136, 10.07438874302542202008744128998
