L(s) = 1 | + 2-s + 4-s + 8-s − 3·9-s − 4·11-s + 4.24·13-s + 16-s − 4.24·17-s − 3·18-s − 5.65·19-s − 4·22-s + 4.24·26-s − 4·29-s + 5.65·31-s + 32-s − 4.24·34-s − 3·36-s − 6·37-s − 5.65·38-s + 1.41·41-s − 12·43-s − 4·44-s + 4.24·52-s − 12·53-s − 4·58-s + 11.3·59-s − 7.07·61-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.353·8-s − 9-s − 1.20·11-s + 1.17·13-s + 0.250·16-s − 1.02·17-s − 0.707·18-s − 1.29·19-s − 0.852·22-s + 0.832·26-s − 0.742·29-s + 1.01·31-s + 0.176·32-s − 0.727·34-s − 0.5·36-s − 0.986·37-s − 0.917·38-s + 0.220·41-s − 1.82·43-s − 0.603·44-s + 0.588·52-s − 1.64·53-s − 0.525·58-s + 1.47·59-s − 0.905·61-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 | \( 1 + 3T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 4.24T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−8.404936149144031677513775353019, −7.951527539037869539182624015648, −6.70412314992962757671132540921, −6.21616518601793966339405085165, −5.37306626198236309732120880153, −4.63838188154805417797408088974, −3.64122797614887584500242126553, −2.79079397579378826809843370943, −1.88171923399790662800702456563, 0,
1.88171923399790662800702456563, 2.79079397579378826809843370943, 3.64122797614887584500242126553, 4.63838188154805417797408088974, 5.37306626198236309732120880153, 6.21616518601793966339405085165, 6.70412314992962757671132540921, 7.951527539037869539182624015648, 8.404936149144031677513775353019