| L(s) = 1 | − 2.37·3-s − 3.37·7-s + 2.62·9-s + 1.37·11-s − 13-s − 1.37·17-s − 4·19-s + 8·21-s − 4.37·23-s + 0.883·27-s − 5.74·29-s + 3.37·31-s − 3.25·33-s + 10·37-s + 2.37·39-s + 8.74·41-s − 3.62·43-s − 4.62·47-s + 4.37·49-s + 3.25·51-s + 11.7·53-s + 9.48·57-s − 1.37·59-s + 1.74·61-s − 8.86·63-s + 8.11·67-s + 10.3·69-s + ⋯ |
| L(s) = 1 | − 1.36·3-s − 1.27·7-s + 0.875·9-s + 0.413·11-s − 0.277·13-s − 0.332·17-s − 0.917·19-s + 1.74·21-s − 0.911·23-s + 0.169·27-s − 1.06·29-s + 0.605·31-s − 0.566·33-s + 1.64·37-s + 0.379·39-s + 1.36·41-s − 0.553·43-s − 0.675·47-s + 0.624·49-s + 0.455·51-s + 1.61·53-s + 1.25·57-s − 0.178·59-s + 0.223·61-s − 1.11·63-s + 0.991·67-s + 1.24·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5891430792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5891430792\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 4.37T + 23T^{2} \) |
| 29 | \( 1 + 5.74T + 29T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 8.74T + 41T^{2} \) |
| 43 | \( 1 + 3.62T + 43T^{2} \) |
| 47 | \( 1 + 4.62T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 1.37T + 59T^{2} \) |
| 61 | \( 1 - 1.74T + 61T^{2} \) |
| 67 | \( 1 - 8.11T + 67T^{2} \) |
| 71 | \( 1 - 3.25T + 71T^{2} \) |
| 73 | \( 1 - 6.74T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 3.48T + 97T^{2} \) |
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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
−9.803852665033002191839170757609, −9.067061341185910541499336976269, −7.910586482311093367027306889450, −6.81927882817439744560330844134, −6.28961827098851955481617228238, −5.71580695280232513698833162249, −4.59076810132731954598289640016, −3.72770722367632266719147808976, −2.34854156975151481598867064963, −0.57761190429443533130309598278,
0.57761190429443533130309598278, 2.34854156975151481598867064963, 3.72770722367632266719147808976, 4.59076810132731954598289640016, 5.71580695280232513698833162249, 6.28961827098851955481617228238, 6.81927882817439744560330844134, 7.910586482311093367027306889450, 9.067061341185910541499336976269, 9.803852665033002191839170757609
