| L(s) = 1 | + 2·2-s + 2·4-s + 5-s + 2·10-s − 11-s + 6·13-s − 4·16-s − 7·17-s + 5·19-s + 2·20-s − 2·22-s + 23-s + 25-s + 12·26-s + 5·29-s + 8·31-s − 8·32-s − 14·34-s − 2·37-s + 10·38-s + 12·41-s − 11·43-s − 2·44-s + 2·46-s + 8·47-s + 2·50-s + 12·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s + 0.632·10-s − 0.301·11-s + 1.66·13-s − 16-s − 1.69·17-s + 1.14·19-s + 0.447·20-s − 0.426·22-s + 0.208·23-s + 1/5·25-s + 2.35·26-s + 0.928·29-s + 1.43·31-s − 1.41·32-s − 2.40·34-s − 0.328·37-s + 1.62·38-s + 1.87·41-s − 1.67·43-s − 0.301·44-s + 0.294·46-s + 1.16·47-s + 0.282·50-s + 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24255 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24255 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.790233978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.790233978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.53074775586860, −14.75718266636684, −14.13265664733843, −13.62944475217430, −13.34167069543993, −13.10881526765570, −12.14940646825382, −11.81957208613561, −11.19563167056848, −10.68786420389415, −10.12692847083581, −9.185971841169699, −8.838843198582996, −8.271245669695601, −7.305386193539123, −6.726599473035065, −6.074555141871379, −5.859315249624684, −4.988100098870231, −4.468917485323670, −3.943704838622395, −3.059156215547264, −2.688863532698208, −1.737810788451390, −0.7706018453301360,
0.7706018453301360, 1.737810788451390, 2.688863532698208, 3.059156215547264, 3.943704838622395, 4.468917485323670, 4.988100098870231, 5.859315249624684, 6.074555141871379, 6.726599473035065, 7.305386193539123, 8.271245669695601, 8.838843198582996, 9.185971841169699, 10.12692847083581, 10.68786420389415, 11.19563167056848, 11.81957208613561, 12.14940646825382, 13.10881526765570, 13.34167069543993, 13.62944475217430, 14.13265664733843, 14.75718266636684, 15.53074775586860
