L(s) = 1 | + 5-s + 2·7-s + 6·13-s − 2·17-s − 19-s + 2·23-s + 25-s + 2·29-s + 4·31-s + 2·35-s − 10·37-s + 10·41-s + 6·43-s + 6·47-s − 3·49-s − 6·53-s + 4·59-s + 2·61-s + 6·65-s − 2·67-s − 12·71-s − 6·73-s + 8·79-s + 2·83-s − 2·85-s − 2·89-s + 12·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s + 1.66·13-s − 0.485·17-s − 0.229·19-s + 0.417·23-s + 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.338·35-s − 1.64·37-s + 1.56·41-s + 0.914·43-s + 0.875·47-s − 3/7·49-s − 0.824·53-s + 0.520·59-s + 0.256·61-s + 0.744·65-s − 0.244·67-s − 1.42·71-s − 0.702·73-s + 0.900·79-s + 0.219·83-s − 0.216·85-s − 0.211·89-s + 1.25·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.513441668\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.513441668\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.694851999000613735781495288463, −7.975642425701524882217327634210, −7.08407840488825294244459268304, −6.26185163564153338558652482863, −5.68990131122318397230996089050, −4.74182762688480039947768216585, −4.01886771733175904580642463706, −2.99293759266190349659968382475, −1.91670462040002196308989838706, −1.01289246869200414139939669709,
1.01289246869200414139939669709, 1.91670462040002196308989838706, 2.99293759266190349659968382475, 4.01886771733175904580642463706, 4.74182762688480039947768216585, 5.68990131122318397230996089050, 6.26185163564153338558652482863, 7.08407840488825294244459268304, 7.975642425701524882217327634210, 8.694851999000613735781495288463