| L(s) = 1 | + 2·3-s − 2·5-s − 9-s + 6·11-s − 6·13-s − 4·15-s + 2·17-s + 12·19-s − 12·23-s + 3·25-s − 6·27-s − 6·29-s + 12·31-s + 12·33-s − 4·37-s − 12·39-s + 4·41-s − 4·43-s + 2·45-s − 6·47-s + 4·51-s − 12·55-s + 24·57-s + 4·59-s + 12·65-s + 8·67-s − 24·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 1/3·9-s + 1.80·11-s − 1.66·13-s − 1.03·15-s + 0.485·17-s + 2.75·19-s − 2.50·23-s + 3/5·25-s − 1.15·27-s − 1.11·29-s + 2.15·31-s + 2.08·33-s − 0.657·37-s − 1.92·39-s + 0.624·41-s − 0.609·43-s + 0.298·45-s − 0.875·47-s + 0.560·51-s − 1.61·55-s + 3.17·57-s + 0.520·59-s + 1.48·65-s + 0.977·67-s − 2.88·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.192341266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.192341266\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 33 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 85 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 104 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 132 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 12 T + 170 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 135 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 18 T + 257 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.441442790812856523883432212045, −8.256325891091565577939245090735, −7.85065505292972220561530559851, −7.82713484789237901041498766749, −7.24035620634240993771593864480, −7.03679416669897953252115658571, −6.44770162826730772498000048947, −6.22515159726651672373737081452, −5.45995213551298798063887261836, −5.43603910847013014003027763175, −4.76043098823771432097158270741, −4.42309441543727187005896293854, −3.88483953880235867248184423100, −3.51597730171840436274601175836, −3.28239995162409282042144765754, −2.92303880235447586509322072964, −2.11145328900084220562022357581, −2.03756948013824219197765158176, −1.07790614661032914947477398655, −0.53499513789367496019032163026,
0.53499513789367496019032163026, 1.07790614661032914947477398655, 2.03756948013824219197765158176, 2.11145328900084220562022357581, 2.92303880235447586509322072964, 3.28239995162409282042144765754, 3.51597730171840436274601175836, 3.88483953880235867248184423100, 4.42309441543727187005896293854, 4.76043098823771432097158270741, 5.43603910847013014003027763175, 5.45995213551298798063887261836, 6.22515159726651672373737081452, 6.44770162826730772498000048947, 7.03679416669897953252115658571, 7.24035620634240993771593864480, 7.82713484789237901041498766749, 7.85065505292972220561530559851, 8.256325891091565577939245090735, 8.441442790812856523883432212045
