| L(s) = 1 | − 4-s + 4·5-s − 9-s − 12·11-s + 16-s − 4·19-s − 4·20-s + 11·25-s + 20·29-s + 8·31-s + 36-s + 20·41-s + 12·44-s − 4·45-s − 2·49-s − 48·55-s + 12·59-s − 12·61-s − 64-s + 4·76-s + 16·79-s + 4·80-s + 81-s − 28·89-s − 16·95-s + 12·99-s − 11·100-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.78·5-s − 1/3·9-s − 3.61·11-s + 1/4·16-s − 0.917·19-s − 0.894·20-s + 11/5·25-s + 3.71·29-s + 1.43·31-s + 1/6·36-s + 3.12·41-s + 1.80·44-s − 0.596·45-s − 2/7·49-s − 6.47·55-s + 1.56·59-s − 1.53·61-s − 1/8·64-s + 0.458·76-s + 1.80·79-s + 0.447·80-s + 1/9·81-s − 2.96·89-s − 1.64·95-s + 1.20·99-s − 1.09·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.536714304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.536714304\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−11.31061088222319326466860001190, −10.86734732684769604164265852953, −10.30856573774620820099580222933, −10.22552635344798454666712342983, −10.07638540664856320547653322637, −9.288432874535801161794638133967, −8.765275334072235731112511737634, −8.285840030009203778344773091551, −7.985260234650288819027843508852, −7.52454283029486581964693688358, −6.46155324443946149085914893388, −6.39222785526780794129104170484, −5.54263794546864161224620363853, −5.41588528894363017704994216290, −4.65374198101652407775169749798, −4.53774726856705535325399598556, −2.83505776121107850648473865027, −2.73768973710073429688874154037, −2.32772330788666862645110641656, −0.826212916009689652085544682030,
0.826212916009689652085544682030, 2.32772330788666862645110641656, 2.73768973710073429688874154037, 2.83505776121107850648473865027, 4.53774726856705535325399598556, 4.65374198101652407775169749798, 5.41588528894363017704994216290, 5.54263794546864161224620363853, 6.39222785526780794129104170484, 6.46155324443946149085914893388, 7.52454283029486581964693688358, 7.985260234650288819027843508852, 8.285840030009203778344773091551, 8.765275334072235731112511737634, 9.288432874535801161794638133967, 10.07638540664856320547653322637, 10.22552635344798454666712342983, 10.30856573774620820099580222933, 10.86734732684769604164265852953, 11.31061088222319326466860001190
