| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s − 11-s − 2·13-s + 16-s + 17-s + 3·19-s − 2·20-s + 22-s + 23-s − 25-s + 2·26-s + 29-s + 2·31-s − 32-s − 34-s − 5·37-s − 3·38-s + 2·40-s − 10·41-s + 43-s − 44-s − 46-s + 7·47-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 − 0.554·13-s + 1/4·16-s + 0.242·17-s + 0.688·19-s − 0.447·20-s + 0.213·22-s + 0.208·23-s − 1/5·25-s + 0.392·26-s + 0.185·29-s + 0.359·31-s − 0.176·32-s − 0.171·34-s − 0.821·37-s − 0.486·38-s + 0.316·40-s − 1.56·41-s + 0.152·43-s − 0.150·44-s − 0.147·46-s + 1.02·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7757359258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7757359258\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.85747770802078815545372392454, −7.07460183022210449527023211966, −6.65765666986493420660734961996, −5.56720138691284046709681890077, −5.02902822390966345594923202582, −4.07525073211865380356777462168, −3.34421680996131307135244884602, −2.59569625731930446489042912023, −1.56672594819181568856264595472, −0.47141604315454646824999418909,
0.47141604315454646824999418909, 1.56672594819181568856264595472, 2.59569625731930446489042912023, 3.34421680996131307135244884602, 4.07525073211865380356777462168, 5.02902822390966345594923202582, 5.56720138691284046709681890077, 6.65765666986493420660734961996, 7.07460183022210449527023211966, 7.85747770802078815545372392454
