| L(s) = 1 | − 3-s − 3.48·5-s + 2.65·7-s + 9-s + 0.483·11-s − 3.48·13-s + 3.48·15-s − 2.61·17-s + 6.78·19-s − 2.65·21-s + 3.61·23-s + 7.13·25-s − 27-s − 7.78·29-s + 31-s − 0.483·33-s − 9.23·35-s + 0.168·37-s + 3.48·39-s − 41-s + 3.13·43-s − 3.48·45-s − 7.45·47-s + 0.0327·49-s + 2.61·51-s + 0.271·53-s − 1.68·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.55·5-s + 1.00·7-s + 0.333·9-s + 0.145·11-s − 0.966·13-s + 0.899·15-s − 0.635·17-s + 1.55·19-s − 0.578·21-s + 0.754·23-s + 1.42·25-s − 0.192·27-s − 1.44·29-s + 0.179·31-s − 0.0841·33-s − 1.56·35-s + 0.0276·37-s + 0.557·39-s − 0.156·41-s + 0.478·43-s − 0.519·45-s − 1.08·47-s + 0.00468·49-s + 0.366·51-s + 0.0372·53-s − 0.227·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9947423719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9947423719\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 + 3.48T + 5T^{2} \) |
| 7 | \( 1 - 2.65T + 7T^{2} \) |
| 11 | \( 1 - 0.483T + 11T^{2} \) |
| 13 | \( 1 + 3.48T + 13T^{2} \) |
| 17 | \( 1 + 2.61T + 17T^{2} \) |
| 19 | \( 1 - 6.78T + 19T^{2} \) |
| 23 | \( 1 - 3.61T + 23T^{2} \) |
| 29 | \( 1 + 7.78T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 0.168T + 37T^{2} \) |
| 43 | \( 1 - 3.13T + 43T^{2} \) |
| 47 | \( 1 + 7.45T + 47T^{2} \) |
| 53 | \( 1 - 0.271T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 6.43T + 61T^{2} \) |
| 67 | \( 1 - 4.13T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 - 6.27T + 79T^{2} \) |
| 83 | \( 1 + 7.75T + 83T^{2} \) |
| 89 | \( 1 - 9.16T + 89T^{2} \) |
| 97 | \( 1 + 6.31T + 97T^{2} \) |
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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
−7.75552295769603510435942689373, −7.24418520252507505888959176941, −6.74157921851963846271631421349, −5.45691043860441337687622691147, −5.06093265031799245851133694469, −4.35019968433148809678690745741, −3.70740946728901599501245465273, −2.77589198069487787186247267130, −1.58016541005128420323072690690, −0.52752625580327931368481962758,
0.52752625580327931368481962758, 1.58016541005128420323072690690, 2.77589198069487787186247267130, 3.70740946728901599501245465273, 4.35019968433148809678690745741, 5.06093265031799245851133694469, 5.45691043860441337687622691147, 6.74157921851963846271631421349, 7.24418520252507505888959176941, 7.75552295769603510435942689373
