| L(s) = 1 | − 3-s + 4·5-s + 2·7-s + 9-s + 2·13-s − 4·15-s − 4·17-s − 6·19-s − 2·21-s + 23-s + 11·25-s − 27-s + 10·29-s + 4·31-s + 8·35-s − 2·37-s − 2·39-s − 6·41-s − 6·43-s + 4·45-s + 8·47-s − 3·49-s + 4·51-s + 8·53-s + 6·57-s − 4·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 0.755·7-s + 1/3·9-s + 0.554·13-s − 1.03·15-s − 0.970·17-s − 1.37·19-s − 0.436·21-s + 0.208·23-s + 11/5·25-s − 0.192·27-s + 1.85·29-s + 0.718·31-s + 1.35·35-s − 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.914·43-s + 0.596·45-s + 1.16·47-s − 3/7·49-s + 0.560·51-s + 1.09·53-s + 0.794·57-s − 0.520·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.751745938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.751745938\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−10.53556902493542170069550644863, −10.22960943101205098974783593913, −8.991042625482035313649647180926, −8.383859171745236175156303525526, −6.72800255281106959324302173868, −6.28265550327556548171712875577, −5.25568786868327562102422643669, −4.44750290738108496231114926922, −2.50490389789723233064355188295, −1.45003740549925543895839662567,
1.45003740549925543895839662567, 2.50490389789723233064355188295, 4.44750290738108496231114926922, 5.25568786868327562102422643669, 6.28265550327556548171712875577, 6.72800255281106959324302173868, 8.383859171745236175156303525526, 8.991042625482035313649647180926, 10.22960943101205098974783593913, 10.53556902493542170069550644863
