| L(s) = 1 | − 2-s + 4-s − 5-s + 4·7-s − 8-s + 10-s − 4·14-s + 16-s − 6·17-s + 4·19-s − 20-s + 25-s + 4·28-s + 6·29-s − 8·31-s − 32-s + 6·34-s − 4·35-s − 2·37-s − 4·38-s + 40-s − 6·41-s − 4·43-s + 9·49-s − 50-s + 6·53-s − 4·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s − 0.353·8-s + 0.316·10-s − 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.223·20-s + 1/5·25-s + 0.755·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s − 0.676·35-s − 0.328·37-s − 0.648·38-s + 0.158·40-s − 0.937·41-s − 0.609·43-s + 9/7·49-s − 0.141·50-s + 0.824·53-s − 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.483929521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.483929521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.14053518750338, −15.39723039489250, −15.09135062427684, −14.52442489488041, −13.79591534155081, −13.42119682591787, −12.42972354814501, −11.93921496590551, −11.42906883266012, −10.94934921078667, −10.53031924333978, −9.738650478149809, −8.926493275875832, −8.640245920608370, −7.982488374652523, −7.456789451853039, −6.903367297060751, −6.172439065914511, −5.170331644423573, −4.838630900354788, −3.993192800684855, −3.166441539297452, −2.164018576384607, −1.611148412917737, −0.6020963377837499,
0.6020963377837499, 1.611148412917737, 2.164018576384607, 3.166441539297452, 3.993192800684855, 4.838630900354788, 5.170331644423573, 6.172439065914511, 6.903367297060751, 7.456789451853039, 7.982488374652523, 8.640245920608370, 8.926493275875832, 9.738650478149809, 10.53031924333978, 10.94934921078667, 11.42906883266012, 11.93921496590551, 12.42972354814501, 13.42119682591787, 13.79591534155081, 14.52442489488041, 15.09135062427684, 15.39723039489250, 16.14053518750338
