| L(s) = 1 | + 2-s + 4-s + 5-s + 3·7-s + 8-s + 10-s + 3·11-s + 3·14-s + 16-s + 3·19-s + 20-s + 3·22-s + 4·23-s + 25-s + 3·28-s + 4·29-s + 6·31-s + 32-s + 3·35-s + 9·37-s + 3·38-s + 40-s + 10·41-s − 10·43-s + 3·44-s + 4·46-s + 3·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.13·7-s + 0.353·8-s + 0.316·10-s + 0.904·11-s + 0.801·14-s + 1/4·16-s + 0.688·19-s + 0.223·20-s + 0.639·22-s + 0.834·23-s + 1/5·25-s + 0.566·28-s + 0.742·29-s + 1.07·31-s + 0.176·32-s + 0.507·35-s + 1.47·37-s + 0.486·38-s + 0.158·40-s + 1.56·41-s − 1.52·43-s + 0.452·44-s + 0.589·46-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.750437118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.750437118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.91784194159383, −15.32766571390859, −14.79433461236084, −14.29780469036011, −13.93483564752400, −13.38359904484086, −12.68700125813138, −12.10107438115249, −11.55778952198922, −11.12532650508156, −10.56032639308529, −9.716790646258137, −9.269276644393708, −8.483319493193100, −7.858166793784486, −7.308270209418495, −6.424280858559674, −6.112238282834809, −5.203792932676810, −4.681323524915566, −4.215192872650559, −3.178068573824337, −2.596699275645133, −1.553778257242763, −1.065923497934192,
1.065923497934192, 1.553778257242763, 2.596699275645133, 3.178068573824337, 4.215192872650559, 4.681323524915566, 5.203792932676810, 6.112238282834809, 6.424280858559674, 7.308270209418495, 7.858166793784486, 8.483319493193100, 9.269276644393708, 9.716790646258137, 10.56032639308529, 11.12532650508156, 11.55778952198922, 12.10107438115249, 12.68700125813138, 13.38359904484086, 13.93483564752400, 14.29780469036011, 14.79433461236084, 15.32766571390859, 15.91784194159383
