| L(s) = 1 | − 84·3-s − 456·7-s + 4.86e3·9-s + 2.52e3·11-s + 1.07e4·13-s + 1.11e4·17-s − 4.12e3·19-s + 3.83e4·21-s + 8.17e4·23-s − 2.25e5·27-s + 9.97e4·29-s + 4.04e4·31-s − 2.12e5·33-s + 4.19e5·37-s − 9.05e5·39-s + 1.41e5·41-s − 6.90e5·43-s − 6.82e5·47-s − 6.15e5·49-s − 9.36e5·51-s − 1.81e6·53-s + 3.46e5·57-s + 9.66e5·59-s + 1.88e6·61-s − 2.22e6·63-s + 2.96e6·67-s − 6.86e6·69-s + ⋯ |
| L(s) = 1 | − 1.79·3-s − 0.502·7-s + 2.22·9-s + 0.571·11-s + 1.36·13-s + 0.550·17-s − 0.137·19-s + 0.902·21-s + 1.40·23-s − 2.20·27-s + 0.759·29-s + 0.244·31-s − 1.02·33-s + 1.36·37-s − 2.44·39-s + 0.320·41-s − 1.32·43-s − 0.958·47-s − 0.747·49-s − 0.988·51-s − 1.67·53-s + 0.247·57-s + 0.612·59-s + 1.06·61-s − 1.11·63-s + 1.20·67-s − 2.51·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.277071747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.277071747\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 28 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 456 T + p^{7} T^{2} \) |
| 11 | \( 1 - 2524 T + p^{7} T^{2} \) |
| 13 | \( 1 - 10778 T + p^{7} T^{2} \) |
| 17 | \( 1 - 11150 T + p^{7} T^{2} \) |
| 19 | \( 1 + 4124 T + p^{7} T^{2} \) |
| 23 | \( 1 - 81704 T + p^{7} T^{2} \) |
| 29 | \( 1 - 99798 T + p^{7} T^{2} \) |
| 31 | \( 1 - 40480 T + p^{7} T^{2} \) |
| 37 | \( 1 - 419442 T + p^{7} T^{2} \) |
| 41 | \( 1 - 141402 T + p^{7} T^{2} \) |
| 43 | \( 1 + 690428 T + p^{7} T^{2} \) |
| 47 | \( 1 + 682032 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1813118 T + p^{7} T^{2} \) |
| 59 | \( 1 - 966028 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1887670 T + p^{7} T^{2} \) |
| 67 | \( 1 - 2965868 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2548232 T + p^{7} T^{2} \) |
| 73 | \( 1 - 1680326 T + p^{7} T^{2} \) |
| 79 | \( 1 + 4038064 T + p^{7} T^{2} \) |
| 83 | \( 1 + 5385764 T + p^{7} T^{2} \) |
| 89 | \( 1 + 6473046 T + p^{7} T^{2} \) |
| 97 | \( 1 - 6065758 T + p^{7} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.26393268541008872472650262649, −9.473416962634956213910735342435, −8.196530631723605786342524747566, −6.76065230888507318881085890229, −6.39410610801494195288238919634, −5.44290996066810598066460142798, −4.48785841711463988509533844542, −3.33894766280170839128428813501, −1.36963623808592972456577269952, −0.65367313124880705666804663317,
0.65367313124880705666804663317, 1.36963623808592972456577269952, 3.33894766280170839128428813501, 4.48785841711463988509533844542, 5.44290996066810598066460142798, 6.39410610801494195288238919634, 6.76065230888507318881085890229, 8.196530631723605786342524747566, 9.473416962634956213910735342435, 10.26393268541008872472650262649
