| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s − 11-s + 13-s + 16-s + 17-s + 8·19-s + 2·20-s + 22-s + 4·23-s − 25-s − 26-s − 6·29-s − 4·31-s − 32-s − 34-s − 6·37-s − 8·38-s − 2·40-s + 6·41-s + 12·43-s − 44-s − 4·46-s − 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s − 0.301·11-s + 0.277·13-s + 1/4·16-s + 0.242·17-s + 1.83·19-s + 0.447·20-s + 0.213·22-s + 0.834·23-s − 1/5·25-s − 0.196·26-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.171·34-s − 0.986·37-s − 1.29·38-s − 0.316·40-s + 0.937·41-s + 1.82·43-s − 0.150·44-s − 0.589·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43758 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43758 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.013396500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.013396500\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.56886629616435, −14.24039396315483, −13.64023326945026, −13.13304959932816, −12.66704946513430, −12.01450422302624, −11.42736986367832, −10.90613952908956, −10.55005025194665, −9.689819193865175, −9.454178708100571, −9.170606828978754, −8.289052799388903, −7.767775064376316, −7.250952871919228, −6.764020147399893, −5.880110036218543, −5.595812104383902, −5.076398985667189, −4.128648110626256, −3.264419767364599, −2.876354079199929, −1.910803786718398, −1.454047538105243, −0.5826479551707481,
0.5826479551707481, 1.454047538105243, 1.910803786718398, 2.876354079199929, 3.264419767364599, 4.128648110626256, 5.076398985667189, 5.595812104383902, 5.880110036218543, 6.764020147399893, 7.250952871919228, 7.767775064376316, 8.289052799388903, 9.170606828978754, 9.454178708100571, 9.689819193865175, 10.55005025194665, 10.90613952908956, 11.42736986367832, 12.01450422302624, 12.66704946513430, 13.13304959932816, 13.64023326945026, 14.24039396315483, 14.56886629616435
