L(s) = 1 | + (−0.587 + 0.809i)5-s + (−0.951 − 0.309i)9-s + (−1.76 + 0.896i)13-s + (−0.896 + 0.142i)17-s + (−0.309 − 0.951i)25-s + (−1.11 − 1.53i)29-s + (0.809 + 1.58i)37-s + (−0.363 + 1.11i)41-s + (0.809 − 0.587i)45-s + i·49-s + (−0.309 − 0.0489i)53-s + (−0.363 − 1.11i)61-s + (0.309 − 1.95i)65-s + (0.142 − 0.278i)73-s + (0.809 + 0.587i)81-s + ⋯ |
L(s) = 1 | + (−0.587 + 0.809i)5-s + (−0.951 − 0.309i)9-s + (−1.76 + 0.896i)13-s + (−0.896 + 0.142i)17-s + (−0.309 − 0.951i)25-s + (−1.11 − 1.53i)29-s + (0.809 + 1.58i)37-s + (−0.363 + 1.11i)41-s + (0.809 − 0.587i)45-s + i·49-s + (−0.309 − 0.0489i)53-s + (−0.363 − 1.11i)61-s + (0.309 − 1.95i)65-s + (0.142 − 0.278i)73-s + (0.809 + 0.587i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2680982497\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2680982497\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.587 - 0.809i)T \) |
good | 3 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (1.76 - 0.896i)T + (0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (0.896 - 0.142i)T + (0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 29 | \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 1.58i)T + (-0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (0.309 + 0.0489i)T + (0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.142 + 0.278i)T + (-0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.278 - 1.76i)T + (-0.951 - 0.309i)T^{2} \) |
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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
−9.788527731974733153929059874014, −9.350997963090286875551756062799, −8.217535631359338123247114988506, −7.62584910515695821944884708007, −6.70461946255714347425846606448, −6.15462438214408811283853939027, −4.88325280003660755778165618259, −4.12762036978884540076924463446, −2.96359690191895761035611322186, −2.23430649740132193869229859567,
0.19013571521074047768085060465, 2.11153614819866869293143234843, 3.14431667478551677439917547126, 4.29238463739474150105366897418, 5.20158125612531101956329560133, 5.63780440767012557735951449441, 7.14052166063600575199750817760, 7.57907129566504486210756755169, 8.560697797309119818651870940566, 9.048708383360608740957749958909