| L(s) = 1 | + 3-s − 3.41·7-s + 9-s + 0.427·11-s − 3.75·13-s − 4.57·17-s − 2.72·19-s − 3.41·21-s − 7.38·23-s + 27-s + 3.93·29-s − 31-s + 0.427·33-s + 2.25·37-s − 3.75·39-s + 11.1·41-s − 6.59·43-s + 3.54·47-s + 4.63·49-s − 4.57·51-s + 6.83·53-s − 2.72·57-s − 10.2·59-s + 13.8·61-s − 3.41·63-s − 10.5·67-s − 7.38·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.28·7-s + 0.333·9-s + 0.129·11-s − 1.04·13-s − 1.10·17-s − 0.624·19-s − 0.744·21-s − 1.54·23-s + 0.192·27-s + 0.730·29-s − 0.179·31-s + 0.0744·33-s + 0.371·37-s − 0.601·39-s + 1.74·41-s − 1.00·43-s + 0.516·47-s + 0.661·49-s − 0.640·51-s + 0.938·53-s − 0.360·57-s − 1.34·59-s + 1.76·61-s − 0.429·63-s − 1.28·67-s − 0.889·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.306256421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.306256421\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 - 0.427T + 11T^{2} \) |
| 13 | \( 1 + 3.75T + 13T^{2} \) |
| 17 | \( 1 + 4.57T + 17T^{2} \) |
| 19 | \( 1 + 2.72T + 19T^{2} \) |
| 23 | \( 1 + 7.38T + 23T^{2} \) |
| 29 | \( 1 - 3.93T + 29T^{2} \) |
| 37 | \( 1 - 2.25T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 6.59T + 43T^{2} \) |
| 47 | \( 1 - 3.54T + 47T^{2} \) |
| 53 | \( 1 - 6.83T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 6.54T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 2.33T + 79T^{2} \) |
| 83 | \( 1 - 3.25T + 83T^{2} \) |
| 89 | \( 1 - 3.11T + 89T^{2} \) |
| 97 | \( 1 - 6.04T + 97T^{2} \) |
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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
−7.66741119502414616174780082278, −7.03522830985161278365650100088, −6.37730826335989819811747775277, −5.88997487008027626334076595932, −4.73587536406559184069586868625, −4.16615031138799747728164246782, −3.41718859280683236759797546661, −2.53450845590113016845043456282, −2.06708466350576792663160556377, −0.49743694194654731119084453867,
0.49743694194654731119084453867, 2.06708466350576792663160556377, 2.53450845590113016845043456282, 3.41718859280683236759797546661, 4.16615031138799747728164246782, 4.73587536406559184069586868625, 5.88997487008027626334076595932, 6.37730826335989819811747775277, 7.03522830985161278365650100088, 7.66741119502414616174780082278
