| L(s) = 1 | + 1.14·2-s + 1.67·3-s − 0.678·4-s − 3.60·5-s + 1.92·6-s − 0.890·7-s − 3.07·8-s − 0.194·9-s − 4.14·10-s + 3.69·11-s − 1.13·12-s + 4.19·13-s − 1.02·14-s − 6.04·15-s − 2.18·16-s + 6.13·17-s − 0.223·18-s − 1.87·19-s + 2.44·20-s − 1.49·21-s + 4.24·22-s + 2.33·23-s − 5.15·24-s + 8.02·25-s + 4.82·26-s − 5.35·27-s + 0.603·28-s + ⋯ |
| L(s) = 1 | + 0.812·2-s + 0.967·3-s − 0.339·4-s − 1.61·5-s + 0.786·6-s − 0.336·7-s − 1.08·8-s − 0.0647·9-s − 1.31·10-s + 1.11·11-s − 0.328·12-s + 1.16·13-s − 0.273·14-s − 1.56·15-s − 0.545·16-s + 1.48·17-s − 0.0526·18-s − 0.431·19-s + 0.547·20-s − 0.325·21-s + 0.905·22-s + 0.486·23-s − 1.05·24-s + 1.60·25-s + 0.945·26-s − 1.02·27-s + 0.114·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.313708935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.313708935\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 1.14T + 2T^{2} \) |
| 3 | \( 1 - 1.67T + 3T^{2} \) |
| 5 | \( 1 + 3.60T + 5T^{2} \) |
| 7 | \( 1 + 0.890T + 7T^{2} \) |
| 11 | \( 1 - 3.69T + 11T^{2} \) |
| 13 | \( 1 - 4.19T + 13T^{2} \) |
| 17 | \( 1 - 6.13T + 17T^{2} \) |
| 19 | \( 1 + 1.87T + 19T^{2} \) |
| 23 | \( 1 - 2.33T + 23T^{2} \) |
| 29 | \( 1 - 5.17T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 - 1.04T + 37T^{2} \) |
| 41 | \( 1 + 1.49T + 41T^{2} \) |
| 47 | \( 1 - 7.55T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 0.0871T + 59T^{2} \) |
| 61 | \( 1 - 8.99T + 61T^{2} \) |
| 67 | \( 1 - 2.36T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 4.41T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 8.14T + 83T^{2} \) |
| 89 | \( 1 - 1.48T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−9.048417685698067488969809752704, −8.279771766092589916132014888666, −8.051442993151238496418196502001, −6.79898834008614554965835558652, −6.05649209454704019273033162862, −4.87608778067913597359695439976, −3.92125345094180338191828222340, −3.57274579721551175227278645138, −2.91876146010429042884968632218, −0.901598155030073717837489039568,
0.901598155030073717837489039568, 2.91876146010429042884968632218, 3.57274579721551175227278645138, 3.92125345094180338191828222340, 4.87608778067913597359695439976, 6.05649209454704019273033162862, 6.79898834008614554965835558652, 8.051442993151238496418196502001, 8.279771766092589916132014888666, 9.048417685698067488969809752704
