| L(s) = 1 | − 2-s + 1.41·3-s + 4-s − 1.41·6-s − 8-s − 0.999·9-s − 2·11-s + 1.41·12-s + 16-s + 1.41·17-s + 0.999·18-s − 7.07·19-s + 2·22-s + 4·23-s − 1.41·24-s − 5.65·27-s + 2·29-s + 8.48·31-s − 32-s − 2.82·33-s − 1.41·34-s − 0.999·36-s − 10·37-s + 7.07·38-s − 9.89·41-s − 2·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.816·3-s + 0.5·4-s − 0.577·6-s − 0.353·8-s − 0.333·9-s − 0.603·11-s + 0.408·12-s + 0.250·16-s + 0.342·17-s + 0.235·18-s − 1.62·19-s + 0.426·22-s + 0.834·23-s − 0.288·24-s − 1.08·27-s + 0.371·29-s + 1.52·31-s − 0.176·32-s − 0.492·33-s − 0.242·34-s − 0.166·36-s − 1.64·37-s + 1.14·38-s − 1.54·41-s − 0.304·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−8.515948099688136973439282906171, −8.142749811253533413828290341214, −7.17201756060828013185414295338, −6.45997921700750033789536909292, −5.49870623595802656801151655402, −4.50446129371836406425191915992, −3.29999362433018355676864180741, −2.65325475390506882403929399354, −1.64863925254873572140399482410, 0,
1.64863925254873572140399482410, 2.65325475390506882403929399354, 3.29999362433018355676864180741, 4.50446129371836406425191915992, 5.49870623595802656801151655402, 6.45997921700750033789536909292, 7.17201756060828013185414295338, 8.142749811253533413828290341214, 8.515948099688136973439282906171
