L(s) = 1 | + 3i·3-s − 10.3i·5-s + 3.46·7-s − 9·9-s − 55.4i·13-s + 31.1·15-s − 90·17-s − 116i·19-s + 10.3i·21-s + 103.·23-s + 17·25-s − 27i·27-s − 259. i·29-s + 301.·31-s − 36i·35-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.929i·5-s + 0.187·7-s − 0.333·9-s − 1.18i·13-s + 0.536·15-s − 1.28·17-s − 1.40i·19-s + 0.107i·21-s + 0.942·23-s + 0.136·25-s − 0.192i·27-s − 1.66i·29-s + 1.74·31-s − 0.173i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.11368 - 0.854559i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11368 - 0.854559i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
good | 5 | \( 1 + 10.3iT - 125T^{2} \) |
| 7 | \( 1 - 3.46T + 343T^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 + 55.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 90T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 259. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 301.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 34.6iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 54T + 6.89e4T^{2} \) |
| 43 | \( 1 - 20iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 394.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 488. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 324iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 575. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 116iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.10e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 148.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 918T + 7.04e5T^{2} \) |
| 97 | \( 1 - 190T + 9.12e5T^{2} \) |
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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
−11.76358435810635047660206708289, −10.92551721093803234054174924686, −9.825714956966005245400193394728, −8.856170098099650617032604101320, −8.099070847895454143489168010785, −6.57102280490647491079067241780, −5.15376237132896869897160849035, −4.44957338670827216535203170801, −2.74385832664733611989329597194, −0.62707656219436239015030994800,
1.71713062226514208791846130543, 3.14294753958494508186816991050, 4.71862083520080706243362678460, 6.39836567773377264526503343090, 6.91009598775252913242746513116, 8.163115666454852848744559814329, 9.227724822903601550069333941686, 10.53085923004328071077976770390, 11.30049800025477708178733277909, 12.21516992556779888799424565687