| L(s) = 1 | − 3·3-s + 5·5-s − 32·7-s + 9·9-s − 64·11-s − 6·13-s − 15·15-s + 38·17-s + 116·19-s + 96·21-s + 120·23-s + 25·25-s − 27·27-s − 122·29-s − 164·31-s + 192·33-s − 160·35-s + 146·37-s + 18·39-s − 238·41-s + 148·43-s + 45·45-s + 184·47-s + 681·49-s − 114·51-s + 470·53-s − 320·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.72·7-s + 1/3·9-s − 1.75·11-s − 0.128·13-s − 0.258·15-s + 0.542·17-s + 1.40·19-s + 0.997·21-s + 1.08·23-s + 1/5·25-s − 0.192·27-s − 0.781·29-s − 0.950·31-s + 1.01·33-s − 0.772·35-s + 0.648·37-s + 0.0739·39-s − 0.906·41-s + 0.524·43-s + 0.149·45-s + 0.571·47-s + 1.98·49-s − 0.313·51-s + 1.21·53-s − 0.784·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.007880741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.007880741\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 64 T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 38 T + p^{3} T^{2} \) |
| 19 | \( 1 - 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 122 T + p^{3} T^{2} \) |
| 31 | \( 1 + 164 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 + 238 T + p^{3} T^{2} \) |
| 43 | \( 1 - 148 T + p^{3} T^{2} \) |
| 47 | \( 1 - 184 T + p^{3} T^{2} \) |
| 53 | \( 1 - 470 T + p^{3} T^{2} \) |
| 59 | \( 1 - 216 T + p^{3} T^{2} \) |
| 61 | \( 1 - 806 T + p^{3} T^{2} \) |
| 67 | \( 1 - 732 T + p^{3} T^{2} \) |
| 71 | \( 1 + 264 T + p^{3} T^{2} \) |
| 73 | \( 1 + 638 T + p^{3} T^{2} \) |
| 79 | \( 1 + 596 T + p^{3} T^{2} \) |
| 83 | \( 1 - 884 T + p^{3} T^{2} \) |
| 89 | \( 1 - 930 T + p^{3} T^{2} \) |
| 97 | \( 1 - 322 T + p^{3} T^{2} \) |
| 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
−10.35668970936836418797612916055, −9.889568316019013171003285133323, −9.053385898339418168324396746647, −7.58280778777313595514944149740, −6.90473770923352443225930021637, −5.68966439800133785173302732395, −5.26699126829130716731504332369, −3.52077095641476754628294820429, −2.58455005720660603215548408403, −0.62137659025727788515712905276,
0.62137659025727788515712905276, 2.58455005720660603215548408403, 3.52077095641476754628294820429, 5.26699126829130716731504332369, 5.68966439800133785173302732395, 6.90473770923352443225930021637, 7.58280778777313595514944149740, 9.053385898339418168324396746647, 9.889568316019013171003285133323, 10.35668970936836418797612916055
