| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 11-s − 6.43·13-s − 15-s + 5.70·17-s − 3.70·19-s − 21-s − 5.70·23-s + 25-s + 27-s + 2.72·29-s + 6.43·31-s + 33-s + 35-s − 6.43·37-s − 6.43·39-s + 8.43·41-s + 10.6·43-s − 45-s + 2.98·47-s + 49-s + 5.70·51-s + 1.70·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 0.333·9-s + 0.301·11-s − 1.78·13-s − 0.258·15-s + 1.38·17-s − 0.850·19-s − 0.218·21-s − 1.19·23-s + 0.200·25-s + 0.192·27-s + 0.506·29-s + 1.15·31-s + 0.174·33-s + 0.169·35-s − 1.05·37-s − 1.03·39-s + 1.31·41-s + 1.63·43-s − 0.149·45-s + 0.434·47-s + 0.142·49-s + 0.799·51-s + 0.234·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.816324322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.816324322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 6.43T + 13T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 + 3.70T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 - 2.72T + 29T^{2} \) |
| 31 | \( 1 - 6.43T + 31T^{2} \) |
| 37 | \( 1 + 6.43T + 37T^{2} \) |
| 41 | \( 1 - 8.43T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 2.98T + 47T^{2} \) |
| 53 | \( 1 - 1.70T + 53T^{2} \) |
| 59 | \( 1 - 5.27T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 - 6T + 67T^{2} \) |
| 71 | \( 1 - 8.43T + 71T^{2} \) |
| 73 | \( 1 - 7.45T + 73T^{2} \) |
| 79 | \( 1 - 8.43T + 79T^{2} \) |
| 83 | \( 1 + 7.70T + 83T^{2} \) |
| 89 | \( 1 + 4.68T + 89T^{2} \) |
| 97 | \( 1 - 6.72T + 97T^{2} \) |
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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
−8.167989866232953375933895572301, −7.68371952006085042811946245506, −7.03542054972109599102012144992, −6.21761715939343370885285960531, −5.32197595586358315151306177496, −4.42326805158848072507372165535, −3.80297443432421383090803800324, −2.82393371199683080599095073219, −2.16536995773068462950401057248, −0.70994218045725268214819081591,
0.70994218045725268214819081591, 2.16536995773068462950401057248, 2.82393371199683080599095073219, 3.80297443432421383090803800324, 4.42326805158848072507372165535, 5.32197595586358315151306177496, 6.21761715939343370885285960531, 7.03542054972109599102012144992, 7.68371952006085042811946245506, 8.167989866232953375933895572301
