| L(s) = 1 | + 2.64i·3-s − 2.23·5-s + 2.64i·7-s − 4.00·9-s + 5.91i·11-s − 6.70·13-s − 5.91i·15-s − 2.23·17-s − 7.00·21-s + 5.00·25-s − 2.64i·27-s + 9·29-s − 15.6·33-s − 5.91i·35-s − 17.7i·39-s + ⋯ |
| L(s) = 1 | + 1.52i·3-s − 0.999·5-s + 0.999i·7-s − 1.33·9-s + 1.78i·11-s − 1.86·13-s − 1.52i·15-s − 0.542·17-s − 1.52·21-s + 1.00·25-s − 0.509i·27-s + 1.67·29-s − 2.72·33-s − 0.999i·35-s − 2.84i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5834862470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5834862470\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 7 | \( 1 - 2.64iT \) |
| good | 3 | \( 1 - 2.64iT - 3T^{2} \) |
| 11 | \( 1 - 5.91iT - 11T^{2} \) |
| 13 | \( 1 + 6.70T + 13T^{2} \) |
| 17 | \( 1 + 2.23T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 7.93iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 17.7iT - 79T^{2} \) |
| 83 | \( 1 + 15.8iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 6.70T + 97T^{2} \) |
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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
−9.615208921768650113051744506561, −9.093598635463148873091595585694, −8.175073562628869286858504268290, −7.38334242591838988587644988940, −6.56551438996456412925581338732, −5.15893206047329879206021831651, −4.73005667992486787414022693333, −4.27490550891597481926856775567, −3.04173402385292809214238443227, −2.27356560256903757789882756755,
0.24657762640180390870168383636, 0.950240097110930196332915060961, 2.44994795696948686479987527019, 3.29224208465527170212894658937, 4.36879751059803863428705388984, 5.32548294049799234151663033713, 6.57587999769590828545691955695, 6.87496469805989900371504755334, 7.73704122540690016629616163729, 8.120296811376074561005713941347
