L(s) = 1 | + i·3-s − 1.97·7-s − 9-s + 1.43i·11-s − 0.241i·13-s + 7.38·17-s − 3.04i·19-s − 1.97i·21-s + 0.874·23-s − i·27-s − 9.07i·29-s + 7.44·31-s − 1.43·33-s + 8.81i·37-s + 0.241·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.747·7-s − 0.333·9-s + 0.431i·11-s − 0.0669i·13-s + 1.79·17-s − 0.697i·19-s − 0.431i·21-s + 0.182·23-s − 0.192i·27-s − 1.68i·29-s + 1.33·31-s − 0.249·33-s + 1.44i·37-s + 0.0386·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.565404121\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.565404121\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.97T + 7T^{2} \) |
| 11 | \( 1 - 1.43iT - 11T^{2} \) |
| 13 | \( 1 + 0.241iT - 13T^{2} \) |
| 17 | \( 1 - 7.38T + 17T^{2} \) |
| 19 | \( 1 + 3.04iT - 19T^{2} \) |
| 23 | \( 1 - 0.874T + 23T^{2} \) |
| 29 | \( 1 + 9.07iT - 29T^{2} \) |
| 31 | \( 1 - 7.44T + 31T^{2} \) |
| 37 | \( 1 - 8.81iT - 37T^{2} \) |
| 41 | \( 1 + 1.91T + 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 + 3.34T + 47T^{2} \) |
| 53 | \( 1 - 9.20iT - 53T^{2} \) |
| 59 | \( 1 - 6.43iT - 59T^{2} \) |
| 61 | \( 1 - 4.57iT - 61T^{2} \) |
| 67 | \( 1 + 4.86iT - 67T^{2} \) |
| 71 | \( 1 - 8.21T + 71T^{2} \) |
| 73 | \( 1 - 4.12T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 12.3iT - 83T^{2} \) |
| 89 | \( 1 + 8.08T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−9.413722662520442884646391341175, −8.236583267303522604662681528228, −7.75097060839066504988763279111, −6.63092804778861014039228400890, −6.04721345483624107302789060492, −5.06730460642395251775279403973, −4.33817855058237547451316198015, −3.29920575347050513103576820390, −2.65313824206539423346189608023, −1.00762980725013027862436398257,
0.66538688625100692452211937851, 1.86075366450334741100665197784, 3.18349894267803782683417213935, 3.61015482947213805480895221562, 5.09305077765848058590459231430, 5.74255889843023813687761790559, 6.55653250175011892614971210863, 7.24059601176411011924705526820, 8.060714501989324224458491842460, 8.690171198646957032311211709847