| L(s) = 1 | + i·3-s + 3·7-s + 2·9-s + 3i·13-s + 3·17-s − i·19-s + 3i·21-s − 9·23-s + 5·25-s + 5i·27-s − 9i·29-s + 6·31-s − 6i·37-s − 3·39-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.13·7-s + 0.666·9-s + 0.832i·13-s + 0.727·17-s − 0.229i·19-s + 0.654i·21-s − 1.87·23-s + 25-s + 0.962i·27-s − 1.67i·29-s + 1.07·31-s − 0.986i·37-s − 0.480·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.024292831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.024292831\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + iT \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 23 | \( 1 + 9T + 23T^{2} \) |
| 29 | \( 1 + 9iT - 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 - 3iT - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 - 5iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.862685587425394053191256341060, −9.145816530441427860233828752262, −8.105497545734099352261462033095, −7.58651326697501713051468396478, −6.46359765609492926333934285601, −5.51248430504078833539322710571, −4.39078225807291656867764951514, −4.14829853802996598658054914664, −2.52003549943530161976396997584, −1.32624340253520185142858625608,
1.07113267627942141792377710412, 2.04295540264713895441864435695, 3.40579877635572868078992330615, 4.57082014948238083712548335703, 5.35132135944355142848853256403, 6.37254600139472960295505130365, 7.28668043003824428231687611588, 8.040319249155155768203095438866, 8.462742243307877688021409433914, 9.809603195376979946669488949759
