| L(s) = 1 | + 2.87·3-s + 5.26·9-s − 2.08·11-s + 0.695·13-s + 2.15·17-s − 8.21·19-s + 7.33·23-s + 6.51·27-s + 2.18·29-s − 7.88·31-s − 5.98·33-s + 1.78·37-s + 2·39-s + 1.48·41-s + 0.216·43-s + 11.6·47-s + 6.20·51-s + 3.14·53-s − 23.6·57-s + 12.4·59-s − 6.84·61-s + 13.1·67-s + 21.0·69-s + 10.3·71-s + 14.5·73-s + 10.9·79-s + 2.93·81-s + ⋯ |
| L(s) = 1 | + 1.65·3-s + 1.75·9-s − 0.627·11-s + 0.192·13-s + 0.523·17-s − 1.88·19-s + 1.52·23-s + 1.25·27-s + 0.405·29-s − 1.41·31-s − 1.04·33-s + 0.293·37-s + 0.320·39-s + 0.231·41-s + 0.0330·43-s + 1.70·47-s + 0.868·51-s + 0.432·53-s − 3.12·57-s + 1.62·59-s − 0.876·61-s + 1.60·67-s + 2.53·69-s + 1.22·71-s + 1.70·73-s + 1.22·79-s + 0.326·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.020200517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.020200517\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.87T + 3T^{2} \) |
| 11 | \( 1 + 2.08T + 11T^{2} \) |
| 13 | \( 1 - 0.695T + 13T^{2} \) |
| 17 | \( 1 - 2.15T + 17T^{2} \) |
| 19 | \( 1 + 8.21T + 19T^{2} \) |
| 23 | \( 1 - 7.33T + 23T^{2} \) |
| 29 | \( 1 - 2.18T + 29T^{2} \) |
| 31 | \( 1 + 7.88T + 31T^{2} \) |
| 37 | \( 1 - 1.78T + 37T^{2} \) |
| 41 | \( 1 - 1.48T + 41T^{2} \) |
| 43 | \( 1 - 0.216T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 3.14T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 6.84T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 1.74T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 - 3.14T + 97T^{2} \) |
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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
−7.85541680928164103863949917854, −7.14663716741198370805464030172, −6.55894738114329045109952980481, −5.54070819053707629216059852841, −4.78351769487244963419208608705, −3.89172126381269512362588973821, −3.45343861638541866727163093397, −2.44067669426493331013866207385, −2.15679729601888611495641418221, −0.874002945697042183210442634176,
0.874002945697042183210442634176, 2.15679729601888611495641418221, 2.44067669426493331013866207385, 3.45343861638541866727163093397, 3.89172126381269512362588973821, 4.78351769487244963419208608705, 5.54070819053707629216059852841, 6.55894738114329045109952980481, 7.14663716741198370805464030172, 7.85541680928164103863949917854
