| L(s) = 1 | + 2.61·2-s + 2.28·3-s + 4.85·4-s + 5.99·6-s + 7.47·8-s + 2.23·9-s − 2.23·11-s + 11.1·12-s − 5.45·13-s + 9.85·16-s − 1.08·17-s + 5.85·18-s − 4.24·19-s − 5.85·22-s + 8.23·23-s + 17.0·24-s − 14.2·26-s − 1.74·27-s − 2.23·29-s + 2.62·31-s + 10.8·32-s − 5.11·33-s − 2.82·34-s + 10.8·36-s − 2.70·37-s − 11.1·38-s − 12.4·39-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 1.32·3-s + 2.42·4-s + 2.44·6-s + 2.64·8-s + 0.745·9-s − 0.674·11-s + 3.20·12-s − 1.51·13-s + 2.46·16-s − 0.262·17-s + 1.37·18-s − 0.973·19-s − 1.24·22-s + 1.71·23-s + 3.49·24-s − 2.79·26-s − 0.336·27-s − 0.415·29-s + 0.470·31-s + 1.91·32-s − 0.890·33-s − 0.485·34-s + 1.80·36-s − 0.445·37-s − 1.80·38-s − 1.99·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.958114979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.958114979\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 3 | \( 1 - 2.28T + 3T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 5.45T + 13T^{2} \) |
| 17 | \( 1 + 1.08T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 - 8.23T + 23T^{2} \) |
| 29 | \( 1 + 2.23T + 29T^{2} \) |
| 31 | \( 1 - 2.62T + 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 - 7.94T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + 9.48T + 47T^{2} \) |
| 53 | \( 1 + 3.70T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 4.70T + 67T^{2} \) |
| 71 | \( 1 - 1.47T + 71T^{2} \) |
| 73 | \( 1 - 6.86T + 73T^{2} \) |
| 79 | \( 1 - 8.70T + 79T^{2} \) |
| 83 | \( 1 - 2.62T + 83T^{2} \) |
| 89 | \( 1 + 4.78T + 89T^{2} \) |
| 97 | \( 1 - 19.5T + 97T^{2} \) |
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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
−9.765334379668373302761045380448, −8.823183249208291155741070299749, −7.75222563683606116415153567748, −7.22571486208130779881219603576, −6.27936240865039804791812502703, −5.12123431615000307054004172364, −4.57268495965836485874787409768, −3.49509575946384560285434856038, −2.69547724599652996019959128126, −2.13992005030042520315325731131,
2.13992005030042520315325731131, 2.69547724599652996019959128126, 3.49509575946384560285434856038, 4.57268495965836485874787409768, 5.12123431615000307054004172364, 6.27936240865039804791812502703, 7.22571486208130779881219603576, 7.75222563683606116415153567748, 8.823183249208291155741070299749, 9.765334379668373302761045380448
