| L(s) = 1 | − i·3-s + 1.34i·7-s − 9-s + 4.69·11-s + 2.38i·13-s − 1.21i·17-s − 1.03·19-s + 1.34·21-s − 6.24i·23-s + i·27-s + 5.59·29-s + 31-s − 4.69i·33-s + 5.41i·37-s + 2.38·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.509i·7-s − 0.333·9-s + 1.41·11-s + 0.661i·13-s − 0.294i·17-s − 0.237·19-s + 0.294·21-s − 1.30i·23-s + 0.192i·27-s + 1.03·29-s + 0.179·31-s − 0.817i·33-s + 0.890i·37-s + 0.381·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.260991742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.260991742\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 1.34iT - 7T^{2} \) |
| 11 | \( 1 - 4.69T + 11T^{2} \) |
| 13 | \( 1 - 2.38iT - 13T^{2} \) |
| 17 | \( 1 + 1.21iT - 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 + 6.24iT - 23T^{2} \) |
| 29 | \( 1 - 5.59T + 29T^{2} \) |
| 37 | \( 1 - 5.41iT - 37T^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 43 | \( 1 + 5.73iT - 43T^{2} \) |
| 47 | \( 1 - 3.48iT - 47T^{2} \) |
| 53 | \( 1 + 2.51iT - 53T^{2} \) |
| 59 | \( 1 + 7.86T + 59T^{2} \) |
| 61 | \( 1 + 5.46T + 61T^{2} \) |
| 67 | \( 1 + 1.41iT - 67T^{2} \) |
| 71 | \( 1 - 3.79T + 71T^{2} \) |
| 73 | \( 1 - 1.41iT - 73T^{2} \) |
| 79 | \( 1 - 1.81T + 79T^{2} \) |
| 83 | \( 1 + 13.2iT - 83T^{2} \) |
| 89 | \( 1 - 2.20T + 89T^{2} \) |
| 97 | \( 1 - 6.69iT - 97T^{2} \) |
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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
−7.63360609406339992054587889900, −6.69847590042513750376360826335, −6.53726247256226328662534122505, −5.81688478551946226918168990734, −4.78175492296438918907599635708, −4.26730805047454881524254216022, −3.29034793549867653236740378444, −2.45382785320837627226646022337, −1.66604880987097711175036501913, −0.71994916544685027654096022272,
0.78205366278657249333629950410, 1.68686025155480652442482674232, 2.88369991148105980478980392632, 3.64262260810589317499088994514, 4.16952673925096077351834377121, 4.89186612812243568232601936303, 5.77861848111714738964245329631, 6.30729032491838283578294028853, 7.10415497828903524098910865068, 7.74627214651632687496804458937
