L(s) = 1 | + i·2-s + 1.55i·3-s − 4-s − 1.01i·5-s − 1.55·6-s − i·8-s + 0.568·9-s + 1.01·10-s + (3.09 + 1.19i)11-s − 1.55i·12-s − 0.167·13-s + 1.58·15-s + 16-s + 2.95·17-s + 0.568i·18-s + 0.311·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.900i·3-s − 0.5·4-s − 0.455i·5-s − 0.636·6-s − 0.353i·8-s + 0.189·9-s + 0.322·10-s + (0.932 + 0.360i)11-s − 0.450i·12-s − 0.0463·13-s + 0.410·15-s + 0.250·16-s + 0.716·17-s + 0.134i·18-s + 0.0715·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.684150703\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.684150703\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-3.09 - 1.19i)T \) |
good | 3 | \( 1 - 1.55iT - 3T^{2} \) |
| 5 | \( 1 + 1.01iT - 5T^{2} \) |
| 13 | \( 1 + 0.167T + 13T^{2} \) |
| 17 | \( 1 - 2.95T + 17T^{2} \) |
| 19 | \( 1 - 0.311T + 19T^{2} \) |
| 23 | \( 1 + 0.475T + 23T^{2} \) |
| 29 | \( 1 - 1.89iT - 29T^{2} \) |
| 31 | \( 1 + 2.54iT - 31T^{2} \) |
| 37 | \( 1 - 6.09T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 - 3.79iT - 47T^{2} \) |
| 53 | \( 1 - 8.42T + 53T^{2} \) |
| 59 | \( 1 - 6.24iT - 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 9.70T + 73T^{2} \) |
| 79 | \( 1 + 7.00iT - 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 6.90iT - 89T^{2} \) |
| 97 | \( 1 + 11.0iT - 97T^{2} \) |
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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
−9.845907736878387326398985302198, −9.354124581153818896057820703371, −8.598760024633027752034982953220, −7.62974815407451013713163831222, −6.77919658392742081257189119105, −5.82200301440321180948950487453, −4.82186345872923114808837655160, −4.28330608145572269565151663612, −3.25066783825006565837053531662, −1.30767683655954480293294324587,
0.923038036608626840282288375545, 1.98487973423324115163027718469, 3.16659272672029409830071678061, 4.09140031784221112396208060259, 5.33362446737946647576886969129, 6.43965984044220764784561086761, 7.05409140125311187094139506455, 8.001328611016199355984901218103, 8.824141667888457746240999467723, 9.771709209507222414931677771456