| L(s) = 1 | − 3-s − 2·5-s + 2·7-s − 9-s + 8·11-s − 3·13-s + 2·15-s + 2·17-s − 8·19-s − 2·21-s + 2·23-s + 3·25-s + 13·29-s + 7·31-s − 8·33-s − 4·35-s + 6·37-s + 3·39-s − 3·41-s + 10·43-s + 2·45-s + 5·47-s + 6·49-s − 2·51-s − 8·53-s − 16·55-s + 8·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.755·7-s − 1/3·9-s + 2.41·11-s − 0.832·13-s + 0.516·15-s + 0.485·17-s − 1.83·19-s − 0.436·21-s + 0.417·23-s + 3/5·25-s + 2.41·29-s + 1.25·31-s − 1.39·33-s − 0.676·35-s + 0.986·37-s + 0.480·39-s − 0.468·41-s + 1.52·43-s + 0.298·45-s + 0.729·47-s + 6/7·49-s − 0.280·51-s − 1.09·53-s − 2.15·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.999961941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.999961941\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 70 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 29 T + 352 T^{2} - 29 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−7.909304284517973614894448412154, −7.890069999890656493233429596824, −7.32357584406751357029683915966, −7.05551878643065207026123010597, −6.48663809919723722378728744220, −6.44216589333450312829571480362, −6.07835927619319091863942860874, −5.84544503805564650869391913762, −4.95437338124771029084920299580, −4.93484620552484266640694938776, −4.44836442739352586412918836602, −4.14850249024400608569949653264, −4.08780815415882570868563975380, −3.31378155882428127757158301864, −2.99241980384728870867482991129, −2.50556784308839949635437618071, −1.96739835241445737859911303643, −1.43911241914330338418995079514, −0.77106967266729678983673732227, −0.64512898965460308642423390595,
0.64512898965460308642423390595, 0.77106967266729678983673732227, 1.43911241914330338418995079514, 1.96739835241445737859911303643, 2.50556784308839949635437618071, 2.99241980384728870867482991129, 3.31378155882428127757158301864, 4.08780815415882570868563975380, 4.14850249024400608569949653264, 4.44836442739352586412918836602, 4.93484620552484266640694938776, 4.95437338124771029084920299580, 5.84544503805564650869391913762, 6.07835927619319091863942860874, 6.44216589333450312829571480362, 6.48663809919723722378728744220, 7.05551878643065207026123010597, 7.32357584406751357029683915966, 7.890069999890656493233429596824, 7.909304284517973614894448412154
