| L(s) = 1 | − 2.44·3-s + 2·5-s − 7-s + 2.99·9-s − 11-s + 4.44·13-s − 4.89·15-s + 4.44·17-s + 4.89·19-s + 2.44·21-s + 8.89·23-s − 25-s − 6.89·29-s + 1.55·31-s + 2.44·33-s − 2·35-s + 4·37-s − 10.8·39-s + 5.34·41-s − 6.89·43-s + 5.99·45-s − 1.55·47-s + 49-s − 10.8·51-s − 9.79·53-s − 2·55-s − 11.9·57-s + ⋯ |
| L(s) = 1 | − 1.41·3-s + 0.894·5-s − 0.377·7-s + 0.999·9-s − 0.301·11-s + 1.23·13-s − 1.26·15-s + 1.07·17-s + 1.12·19-s + 0.534·21-s + 1.85·23-s − 0.200·25-s − 1.28·29-s + 0.278·31-s + 0.426·33-s − 0.338·35-s + 0.657·37-s − 1.74·39-s + 0.835·41-s − 1.05·43-s + 0.894·45-s − 0.226·47-s + 0.142·49-s − 1.52·51-s − 1.34·53-s − 0.269·55-s − 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9790907047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9790907047\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 13 | \( 1 - 4.44T + 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 - 8.89T + 23T^{2} \) |
| 29 | \( 1 + 6.89T + 29T^{2} \) |
| 31 | \( 1 - 1.55T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 5.34T + 41T^{2} \) |
| 43 | \( 1 + 6.89T + 43T^{2} \) |
| 47 | \( 1 + 1.55T + 47T^{2} \) |
| 53 | \( 1 + 9.79T + 53T^{2} \) |
| 59 | \( 1 + 7.34T + 59T^{2} \) |
| 61 | \( 1 + 9.34T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 + 3.10T + 71T^{2} \) |
| 73 | \( 1 - 7.55T + 73T^{2} \) |
| 79 | \( 1 + 1.10T + 79T^{2} \) |
| 83 | \( 1 + 7.10T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−11.48394573060304426569075730966, −10.93268980626317002630002221542, −9.954959392489633817784112235944, −9.177677313425725443745857663632, −7.65766783860508522896836152125, −6.46071100623735997890262068396, −5.75832604746648542077413438749, −5.04231410927294119410876844444, −3.28513107251763483330736154463, −1.19119964338012444166540234020,
1.19119964338012444166540234020, 3.28513107251763483330736154463, 5.04231410927294119410876844444, 5.75832604746648542077413438749, 6.46071100623735997890262068396, 7.65766783860508522896836152125, 9.177677313425725443745857663632, 9.954959392489633817784112235944, 10.93268980626317002630002221542, 11.48394573060304426569075730966
